“Don’t send a boy to do a man’s job/work” means that one should not have someone do what that person cannot do. “Hire a boy to do a man’s business; he will make up his wages in the spoiling of your tools” was cited in 1860.
 
“Never send a boy to do a man’s work” was cited in print in 1874 and “Don’t send a boy to do a man’s job” was cited in print in 1920. “We won’t send a girl to do a woman’s job” was cited in an advertisement for temporary services in 1966.
 
   
TV Tropes
Never Send an X to Do a Y’s Job
A stock phrase, said when one character has to take care of something another tried and failed to do. The most common phrasing is, “Never send a boy to do a man’s job.”
 
Google Books
Optimism:
The Lesson of Ages
By Benjamin Paul Blood
Boston, MA: Bela Marsh
1860
Pg. 92:
Hire a boy to do a man’s business; he will make up his wages in the spoiling of your tools. Though he may do well for a while, some accident will revenge your penuriousness.
 
12 October 1874, The Daily Graphic (New York, NY), pg. 736, col. 4:
Never send a boy to do a man’s work.
   
Google Books
The History of the United States:
Told in One Syllable Words
By Josephine Pollard
New York, NY: McLoughlin Brothers, Publishers
1884
Pg. 31:
It is not well to set a boy to do a man’s work.
 
Google Books
December 1890, The Indicator, pg. 583, col. 1:
I quoted the old saying: “Never send a boy to do a man’s work,” and commented on it in a facetious manner; ...
 
27 June 1920, Springfield (MA) Sunday Republican, pg. 2A, col. 4 ad:
“Don’t send a boy to do a man’s job.”
(Berkshire Motor Co.—ed.)
 
Google Books
9 April 1956, Life magazine, “Man in a Gray Flannel Trap,” pg. 111, col. 2:
The movie’s producers acted on an old maxim Madison Avenue is fond of mouthing, “Don’t send a boy to do a man’s job,” and threw $4 million (including a $1.5 million cast) into Man.
     
Google Books
Dun’s Review and Modern Industry
Volume 88
1966
Pg. 66 ad:
We separate the women from the girls. And that’s what separates us from other temporary services. We won’t send a girl to do a woman’s job.
 
Google Books
The Dictionary of Modern Proverbs
Compiled by Charles C. Doyle, Wolfgang Mieder, and Fred Shapiro
New Haven, CT: Yale University Press
2012
Pg. 25:
A boy cannot do a man’s work.
1904 Charles Richmond Henderson, Introduction to the Study of Dependent, Defective, and Delinquent Classes, 2nd ed. (Boston: D. C. Heath) 180: “Mr. A. Johnson makes a distinction, however, which should be noted. A boy cannot do a man’s work; if he could he would be more than self-supporting, for a man must support several persons by his labor.” DAP 64. In contrast to the (slightly) older “Don’t send a boy to do a man’s job,” the proverb “A boy cannot do a man’s work”—in one of its applications, at least—has more to do with economics than with competence.