'Sitters' of various sorts were much in the news in the early 1930s. Tree-sitters, flagpole sitters and house sitters all made appearances. For example, this concerning a long-staying house guest, from the Barnard Bulletin (New York, New York) of 07 Nov 1930:
As his host remarks, "I've heard of flag-pole sitters and tree sitters, but, by God, this fellow's a house sitter!"
Note that flag-pole sitting, tree sitting and, in this case, house sitting, did not connote care for or attention to, respectively, flag-poles, trees and houses, but rather were practiced as faddish and capricious tests or contests of endurance.
The term 'baby sitters', however, so far as I can discover, was coined by or to describe one arm of the "Hop Light Ladies" in 1933, as mentioned in other answers. The "Hop Light Ladies" were a group of female entrepreneurs who, "before the depression were writers, sculptors, painters, interior decorators and business women". Syndicated stories about these depression-era worthies were edited variously. One version states that a
younger group [of the Hop Light Ladies], to be known as "Baby Sitters," will offer their services to supply reliable women at low cost to couples who wish to leave their youngsters in reliable hands at home while they have an evening out.
The Tennessean (Nashville, Tennessee), 25 May 1933.
The phrasing of the article intimates (but only intimates) that the term was coined by the Hop Light Ladies to describe one of their many money-making endeavors.
'Baby-sitter' did not appear in a vacuum, of course. Influences on its development and adoption worth mentioning, other than the above-attested general prevalence in depression-era US news of 'sitters' of various sorts, include earlier use in the US of the slang term 'baby-herder', along with slang use of the noun 'sit' as a shortening of 'situation'.
The somewhat earlier 'baby-herder' and later 'baby-sitter' are in contrast to the now-historical and derogatory 'baby-farmer' (1867), one "who undertakes the charge of children for a fixed sum" (OED). 'Baby-sitter' may well have been deliberately coined with the idea of avoiding entrepreneurial association with the practice of 'baby-farming'.
Add to these influences that 'sit' (the verb) had long been used with appositive complements to denote "the position or occupation of a person" (OED, sit, v., 7c, attested from around 825).
Of the 'baby-herders' and the noun 'sit' meaning "situation", the following evidence pertains.
The slang 'sit' meaning "situation" appeared first, in 1853. Originally, the term referred to type compositers:
slang (orig. Printing).
= SITUATION n. 6b.
1853 ‘M. Twain’ in Hannibal (Missouri) Jrnl. 8 Sept. 2/1 I shall look out for a sit; for they say there is plenty of work to be had for sober compositors.
Use of the slang 'sit' is further attested in OED through 1914. Also, numerous 'Work Wanted' ads of the following general form appear in US newspapers throughout the late decades of the 1800s and the early decades of the 1900s:
(from the St. Louis Post-Dispatch, Missouri, 20 Sep 1912).
The US slang 'baby herder' does not appear in OED. I found a scattering of appearances of the term in US newspapers from 1875 through 1947. The first, from 1875, makes the connection between 'sit' and 'baby herder':
Many of the miners of the Salmon mountains are wintering in the Lembi valley, some of whom have received temporary "sits" as "baby herders", receiving therefor regular rations of substantial grub.
The Independent-Record (Helena, Montana), 16 Mar 1875.
Further appearances (for example, in 1878, 1887, 1888, 1891, 1894) firmly establish that the intended meaning of 'baby herder' is 'baby sitter', and a 1947 appearance in an article listing some of the contents of The American Thesarus of Slang (Lester V. Berrey and Melvin Van den Burk) provides one definition:
A nurse maid is a baby herder or kindergardener.
*The Winnipeg Tribune (Winnipeg, Manitoba, Canada), 23 Apr 1947.
One Spurious 1914 Appearance of 'Baby-Sitting'
The 1914 first appearance date given unattested by Etymology Online may have been sponsored by an OCR error, as shown under About this Book ("Baby-sitting campaigns pliminary report..." at left) in the Google scan of a 1914 US Government Printing Office monograph titled Baby-Saving Campaigns, produced by the US Department of Labor Children's Bureau.