The saying “drain the swamp” has had a long history in American politics.  The saying originally meant to “drain the swamp” to get rid of malaria-carrying mosquitoes. Victor Berger (1912) and Mother Jones (1913) suggested “draining the swamp” to get rid of capitalism.
   
By April 1970, another “drain the swamp” saying had emerged: “When you’re up to your ass in alligators, it’s easy to forget you came to drain the swamp.” U.S. presidential candidate Ronald Reagan said in March 1976:
 
”“Sometimes, when you are up to your elbows in alligators, it is hard to remember your original objective was to drain the swamp. I think we can drain the swamp. We can take on the Washington system. We can change from remote control to personal control of our lives.”
 
U.S. President Ronald Reagan, in January 1982, looked back on a year in office and reminded himself that ‘‘you’re here to drain the swamp’’ of big government. Washington, D.C., by the Potomac River, often fits the “swamp” saying. In 2001, after terror attacks on the Pentagon and the World Trade Center, Secretary of Defense Donald Rumsfeld hoped to “drain the swamp” that supports terrorism. In October 2006, soon-to-be House Speaker Nancy Pelosi pledged to “drain the swamp of Washington.”
   
     
Wiktionary: drain the swamp when up to one’s neck in alligators
Proverb
When you’re up to your neck in alligators, it’s easy to forget that the initial objective was to drain the swamp.
(idiomatic) When performing a long and complex task, and when you’ve gotten utterly immersed in secondary and tertiary unexpected tangential subtasks, it’s easy to lose sight of the initial objective. This sort of distraction can be particularly problematic if the all-consuming subtask or sub-subtask is not, after all, particularly vital to the original, primary goal, but ends up sucking up time and resources (out of all proportion to its actual importance) only because it seems so urgent.
     
(Oxford English Dictionary)
Chiefly U.S. Politics. to drain the swamp: to rid an institution or society of an entrenched and harmful influence, esp. a source or agent of corruption.
In later use, popularized by the use by Ronald Reagan in quot. 1982. See also quot. 1982   at up to one’s armpits in alligators at alligator n.2 Phrases   from the same speech.
Frequently as part of an extended metaphor; see also when you’re up to your ass in alligators, it’s hard to remember that you’re there to drain the swamp at alligator n.2 Phrases.
1899   Arena June 769   The swamp of bossism and corruption is at last being drained, and the solid ground of honest and popular administration is being regained.
1912   V. L. Berger Broadsides 107   We should have to drain the swamp—change the capitalist system—if we want to get rid of those mosquitos.
1982   R. Reagan in Register (Orange County, Calif.) 21 Jan. a17/1   That is why you’re here and I’m here:..to drain the swamp of over-taxation, over-regulation and runaway inflation that has dangerously eroded our free way of life.
   
Newspapers.com
10 October 1903, Oshkosh (WI) Daily Northwestern, pg. 8, col. 7:
Socialists are not satisfied with killing a few of the mosquitoes which come from the capitalist swamp; they want to drain the swamp.
(Letter from Winfield R. Gaylord, State Organizer, Social Democratic Party—ed.)
   
Google Books
Berger’s Broadsides
By Victor L. Berger
Milwaukee, WI: Social-Democratic Pub. Co.
1912
Pg. 107:
So this is another evil that is inherent in this system.
 
It cannot be avoided any more than malaria in a swampy country. And the speculators are the mosquitos.
 
We should have to drain the swamp — change the capitalist system — if we want to get rid of those mosquitos. 
 
Teddy Roosevelt, by starting a little fire here and there to drive them out, is simply disturbing them.
       
Google Books
Currnet Opinion
1913  
Pg. 20:
The capitalist and striker — both men are all right — only they are sick; they need a remedy; they have been mosquito bitten. Let’s kill the virulent mosquito and then find and drain the swamp in which he breeds.
(Spoken by labor organizer Mary Harris “Mother” Jones—ed.)
 
Google Books
Sex and Common Sense
By Agnes Maude Royden
New York, BY: G.P. Putnam’s Sons
1928
Pg. 51:
Drain the swamp, and you get rid of the malaria, for there is no longer any place for the malaria-bearing mosquito to breed. Drain the swamp of immorality, and you get rid of venereal disease, because there is no longer a place where these diseases can breed.
 
21 July 1960, Panola Watchman (Carthage, TX), “Anderson’s Platform For America,” pg. 5, col. 1: 
Bills to limit the ceiling of the Federal income tax are just swatting flies when we need to drain the swamp.
(Tom Anderson, editor of Farm and Ranch magazine—ed.)
   
19 February 1963, Port Angeles (WA) Evening News, Letters to the editor, pg. 2, col. 2:
Congress is not going to repeal socialistic programs one at a time. Instead of swatting mosquitoes, we need to drain the swamp.
   
2 January 1970, Ames (IA) Daily Tribune, “From My Point of View” by Rod Riggs, pg. 4, cols. 3-4:
After watching me at my desk for a few moments on day last week, Jack Smalling offered me a bit of management advice: “When you’re up to your hips in alligators,” he said, “it’s difficult to remind yourself that the original objective was to drain the swamp.”
 
Google News Archive
20 April 1970, Nevada (MO) Daily Mail, pg. 6, col. 2:
WASHINGTON (AP)—A newsletter issued by the Kansas Cooperative Council includes this observation regarding procrastination over new farm legislation and its importance to farmers:
 
“When a man is up to his shorttail in alligators, he has difficulty reminding himself that his initial objective was to drain the swamp.”
     
Google Books
Blind Man on a Freeway
By William Moore
San Francisco, CA: Jossey-Bass
1971
Pg. 32:
When one is up to his ass in alligators, it is easy to forget that his original objective was to drain the swamp.
 
Google Books
The Realities of Crime and Punishment
By Fred T Wilkinson and Fred DeArmond
Springfield, MO: Mycroft Press
1972
Pg. 249:
There comes a time when “We need to stop swatting mosquitos and drain the swamp.”
   
Google Books
Man, government, and the sea : northern Puget Sound and the strait of Georgia : proceedings of a conference held at Western Washington State College, September 15-September 19, 1975
Edited by James William Scott and Manfred C Vernon
Bellingham, WA: Center for Pacific Northwest Studies, Western Washington State College
1976
Pg. 4:
It’s rather like trying to drain the swamp when you are up to your Adam’s apple in alligators.
     
12 March 1976, Decatur (IL) Review, “Ford, Reagan Philosophically Close Together” by James R. Dickenson (Washington Star), pg. 6, col. 4:
Ronald Reagan was speaking recently to the Marion County Chamber of Commerce in Ocala, Fla., and inevitably his favorite subject, cutting back the federal government, came up.
 
“Sometimes, when you are up to your elbows in alligators, it is hard to remember your original objective was to drain the swamp,” he said. “I think we can drain the swamp. We can take on the Washington system. We can change from remote control to personal control of our lives….”
 
New York (NY) Times
President Marks His First Year
AP
Published: January 21, 1982
WASHINGTON, Jan. 20
President Reagan gathered the hierarchy of his Administration today to toast his first anniversary in office and to remind them that ‘‘you’re here to drain the swamp’’ of big Government.
 
Google News Archive
24 January 1983, Florence (SC) Tiimes-Daily, “Reading Howard’s Mind” (Howard Baker—ed.) by William Safire, pg. 4, col. 5:
After all, if you can’t drain the swamp, you can at least shoot the alligators. (I have to collect more expressions like that; they’re colorful and quotable and add to that down-home, old-shoe image. But wait—do you suppose many environmentalists object to the shooting of alligators?)
 
Google News Archive
24 February 1983, Cape Girandeau (MO) Bulletin-Journal, “Haynesworth revisited” by Patrick J. N=Buchanan, pg. 2, col. 6:
As the Adelman nomination reaches the Senate floor, the ensuing battle should do much to clear the air, to re-identify for a sometimes forgetful White House who its true friends are, and where the Sunshine Republicans reside, and, hopefully, remind the White House—yet again—that they were not sent here to make friends with the alligators, but to drain the swamp.
 
Google News Archive
4 August 1985, Gainesville (FL) Sun, “Reagan lauds Congress in fight with deficit,”
“Sometimes it’s difficult to remember, you didn’t send us to Washington to feed the alligators, you sent us to drain the swamp. We didn’t come to raise your taxes, but to lower them,” the president said.
 
OCLC WorldCat record
Stop chasing aligators and drain the swamp : plus: What love is and isn’t.
Author: A Lynn Scoresby
Publisher: Salt Lake City, Utah : Covenant Communications, Inc, 1990.
Edition/Format: Book : English
 
OCLC WorldCat record
When you’re up to your neck in alligators, it’s hard to remember that the original aim was to drain the swamp: Some lessons from New Zealand health sector reform
Author: L C Fulcher
Edition/Format: Article
Publication: AUSTRALIAN SOCIAL WORK, 47, no. 2, (1994): 47
Database: British Library Serials
     
Google News Archive
11 September 1994, Sumter (SC) The Item, “Flat tax: More than an economic Band-Aid” by George Will (Washington Post), pg. 8A, col. 2:
WASHINGTON—The first thing—make that the only thing—Congress should do before adjourning is pass Rep. Dick Armey’s “Freedom and Fairness Restoration Act.” Congress won’t, any more than a crocodile will drain the swamp that is its habitat.
 
New York (NY) Daily News
PREZ EDIFICE PUT AT 738M
By KATHY KIELY Daily News Washington Bureau
Sunday, June 29th 1997, 2:02AM
WASHINGTON Government downsizing is all the rage, but you’d never know it from the monumental building your tax dollars just put up for several thousand bureaucrats.
 
And of all people, it’s named for President Ronald Reagan, who came to town in 1981 determined to drain the “swamp,” as he called the bloated federal bureaucracy.
 
The 3.1 million-square-foot Ronald Reagan Building and International Trade Center is bigger though lower than the Empire State Building, covering 7.7 acres of prime downtown real estate.
     
CNN.com
Rumsfeld: U.S. must ‘drain the swamp’
Roosevelt carrier group headed to sea
September 19, 2001 Posted: 11:30 AM EDT (1530 GMT)
WASHINGTON (CNN)—As a U.S. aircraft carrier battle group prepared to leave for the Mediterranean Sea, Defense Secretary Donald Rumsfeld said Tuesday the United States hoped to “drain the swamp” that supports terrorism.
   
The Guardian
Drain the swamp and there will be no more mosquitoes
By attacking Iraq, the US will invite a new wave of terrorist attacks
Noam Chomsky
The Guardian, Monday 9 September 2002 01.37 BST
 
New York (NY) Daily News
ONE SWAMPY MESS. If Bush is to pull his second term out of the muck, he must confront pols & lobbyists who squander the public trust
BY MORTIMER B. ZUCKERMAN
Monday, October 10th 2005, 6:51AM
Republicans came to power pledging - in their own words - to drain the swamp of special interests in Washington and restore integrity to the White House and the Congress. Instead, President Bush’s second term is mired in potential scandal and demonstrable incompetence.
 
FOXNEWS.com
Pelosi Says She Would Drain GOP ‘Swamp’
Friday, October 06, 2006
By DAVID ESPO, AP Special Correspondent
WASHINGTON —  Franklin Roosevelt had his first hundred days. House Democratic leader Nancy Pelosi is thinking 100 hours. Time enough, she says, to begin to"drain the swamp"after more than a decade of Republican rule.
 
New York (NY) Times
In a first, it will be Madam Speaker
Published: Sunday, October 8, 2006
WASHINGTON — Nancy Pelosi, who stands to become the first woman to lead the House of Representatives after a strong national showing by her fellow Democrats, said the party is “ready to lead.”
(...)
But during the campaign she also promised a barrage of “discrete deliverables” in the first 100 work hours after the Democrats take control. She said a Democratic-led veto-proof majority would “drain the swamp of Washington,” create a new rules package that would include pay-as-you-go budgeting, adopt immediately the recommendations of the 9/11 commission, increase the minimum wage to $7.25 and broaden the types of stem cell research allowed with federal money. 
   
New York (NY) Times
Editorial
Is the House Swamp Drained Yet?
Published: April 18, 2009
(...)
Speaker Pelosi won her job with a convincing denunciation of the Republican Congress’s “culture of corruption,” plus a ringing promise to “drain the swamp” where House ethical standards festered. She deserves credit for forcing through the new quasi-independent Office of Congressional Ethics to vet and funnel complaints to the sitting ethics committee.
   
Amazon.com
It’s Hard to Drain the Swamp When Yer up to Yer Ears in Alligators! by Rockie Sue Fordham (Paperback - Mar. 26, 2010)
   
The Hill
Hoyer: “Drain the swamp” not my line
By Jared Allen - 07/27/10 01:21 PM ET
House Majority Leader Steny Hoyer (D-Md.) on Tuesday noted that it was Speaker Nancy Pelosi (D-Calif.), not him, who promised to “drain the swamp” of corruption in Washington.
 
Pelosi famously vowed in 2006 to “drain the swamp” that ensnared Republican members and their leaders during the George W. Bush administration.
 
Politifact
Madeleine Albright compares Mussolini, Trump on use of ‘drain the swamp’
By Louis Jacobson on Wednesday, May 9th, 2018 at 9:24 a.m.
(...)
The publication of Albright’s Fascism: A Warning during the Trump era is not coincidental. “Why, this far into the twenty-first century, are we once again talking about fascism?” Albright wrote. “One reason, frankly, is Donald Trump. If we think of fascism as a wound from the past that had almost healed, Trump in the White House was like ripping off the bandage and picking at the scab.”
(...)
That said, the phrase remained rare until about 1970, Popik found, and it didn’t really catch on until President Ronald Reagan used it in 1983 to take a shot at the Washington bureaucracy.
 
Popik suggests that, rather than referring to the Pontine marshes, “drain the swamp” probably came from a folksy aphorism that posits that “when you’re up to your neck in alligators, it’s easy to forget that the initial objective was to drain the swamp.”
 
Through her publisher, Albright’s communications staff agreed with our finding that “drain the swamp” or its Italian-language equivalent was not a major slogan for Mussolini’s government.