“How do you keep/stop an elephant from charging? You take away his credit card!”
 
This joke (along with other elephant jokes) went viral in 1963 and 1964 and is of unknown authorship.
   
 
Google News Archive
8 August 1963, Wilmington (NC) Morning Star,  “Have You Heard?” by Theresa Thomas, pg. 6, col. 1:
The sample joke goes like this: How do you stop an elephant from charging? Take away his credit card.
 
Yes, our reaction was “Ough” too.
   
3 September 1963, Ames (IA) Daily Tribune, “From My Point of VIew” by Rod Riggs, pg. 4, col. 2:
Q Know how to keep an elephant from charging?
A Take away his credit card.
 
23 October 1963, The Garfieldian (Chicago, IL), “Just One Happy Whirl” by Don Madden, pg. 20, col. 1:
Corney Joke Dept….Bill Hosy of Sincere and Co. asked us how to keep an elephant from charging….We, as usual didn’t know….Sed Bill, it’s easy, “cut his charge account.”
 
28 October 1963, Bristol Courier and Levittown Times (Bristol and Levittown, PA), “High School News and Views,” pg. 22, col. 3:
QUES. — How do you keep an elephant from charging?
ANS. — Take away his credit card.
     
30 December 1964, San Diego (CA) Union, “Strange Things Happened In ‘64” by Donald Freeman, Pg. A6, col. 1:
It was a year for wacky jokes which spread like a good-humored plague. (...) How do you stop an elephant from charging? Take away his credit card.
 
21 January 1968, Sunday Herald Traveler (Boston, MA), “Charlie Rice’s Punchbowl,” This Week magazine, pg. 26, col. 3:
QUESTION: How can you stop an elephant from charging?
ANSWER: Just take away his credit card.
 
Google Books
Children’s Riddling
By John Holmes McDowell
Bloomington, IN: Indiana University Press
1979
Pg. 246:
How do you keep an elephant from charging? Take away his credit card. (boy, 8)
Barrick, SFQ 1964 #67
   
Google Books
W. C. Privy’s Original Bathroom Companion
Edited by Erin Barrett and Jack Mingo
New York, NY: St. Martin’s Griffin
2003
Pg. 197:
How do you keep an elephant from charging? Take away his credit cards.
 
Twitter
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‏@DigBeingADad
Q: How do you stop an elephant from charging? A: Take away his credit cards.
3:53 PM - 18 Nov 2014