RaceYouAnyTime's answer covers the emergence of the actual words prescriptivism and prescriptivist quite well, and I have nothing to add to that account. The source phrase prescriptive grammar, however, appears to go back a bit farther than Jesperson's 1933 book, Essentials of English Grammar.
Lee Deighton, The Encyclopedia of Education, volume 3 (1971) cites an article titled "Notes on the Founders of Prescriptive Grammar," which appeared in Manly Anniversary Studies in Language and Literature (University of Chicago Press, 1923) [combined snippets]:
The new understanding of the nature of language and of its constant variation led to increasing concern with the disparity between English as actually used and English as prescribed by authoritarian school rules. One outcome of this concern was attention to the historical background. Hints that the authoritarian rules had an unreal foundation first appeared in J. L. Moore’s article "Tudor-Stuart Views on the Growth Status, and Destiny of the English Language" in 1910. How these hints could be followed appeared in an outline offered by W. F. Bryan in "Notes on the Founders of Prescriptive Grammar" (1923). In 1925, Charles C. Fries demonstrated the historical invalidity of the usual rigid prescriptions about expressions indicating future time in "The Periphrastic Future with Shall and Will in Modern English." Among subsequent influential studies was Sterling Andrus Leonard's The Doctrine of Correctness in English Usage, 1700-1800 (1929).
Bryan's eleven-page article, viewable in its entirety at Hathi Trust, begins as follows:
Complete treatises of English grammar, undertaking to determine proper constructions and usage and thus to fashion public taste, were creations of the later eighteenth century. The first of the prescriptive grammarians whose work contributed greatly to the formation of conventional standards was Robert Lowth, successively Bishop of St. David's, Oxford, and London,, and a distinguished classical and oriental scholar.
And its final paragraph begins with this remark:
The number of illustrations of matters thus debated might of course be very greatly increased; those that have been given [in this article] are merely representative of points about which there was greatest divergence of opinion. Though, in themselves, these illustrations may be of only slight consequence, they have real significance as representing the attitudes of the fathers of critical and prescriptive English grammar.
But those are the only occurrences of prescriptive or any closely related form the word.
A search of Moore's 1910 article (which is actually a book) finds no instances of prescriptive, prescribe, prescribes, or prescribing, and only one match for prescription—that being as part of a quotation from Richard Carew, "The Excellencie of the English Tongue by R.C. of Anthony Esquire to W.C. [William Camden]," in Camden's Remains Concerning Britaine (1637). And a search of Fries's 1925 article finds no matches for any word in the prescriptive family.
Leonard's 1929 book uses prescriptive only once in the running text:
Most prescriptive grammars insist on "it seems to be he," but the contrary ruling is also to be found.
But in his bibliography, under "Additional Sources of Material," Leonard also cites Bryan's article by title.
I think that Bryan's use in 1924 of "prescriptive grammar" and "prescriptive grammarian" mark the starting point of widespread awareness of those terms and (by extension) is the probable source of the later terms prescriptivism and prescriptivist.