I'm going to hazard a guess that the pronunciation is based on two things: widespread uncertainty as to how antisemitic is spelled, and pronunciation conditioning based on similar-sounding words.
Early confusion about how to spell the word 'antisemitic'
With regard to the first issue, I note that the spelling antisemetic is common enough in published (and presumably edited) writings since 1900 to support its own Ngram chart:
The matches underlying this chart include this one from The Advocate: America's Jewish Journal (September 2, 1922):
ANTISEMETIC MOVEMENT IN GERMANY INCREASING
Berlin (J. T. A.)—In spite of an order issued by the Ministry of Education prohibiting the wearing of anti-Semetic designs in schools, the Swastika, which has become the symbol of anti-Semetism, is being worn by students, and even teachers in the schools and universities.
Earlier still is this review of Abraham Cahan, White Terror and the Red, from Book Review Digest, volume 1 (1905):
A story of inner Russia by a member of the Revolutionary party who was forced to flee from Russia to avoid Siberia. The plot concerns the tragic events of a quarter of a century ago, when czar Alexander II. was assassinated by the Nihilists and an antisemetic outbreak followed, but it is the Russia of to-day that we see, drawn from a practical knowledge of the subject.
And 24 years before that, we have this instance from the Rev. Dr. Stern, "The Talmud," in Proceedings of the Literary & Philosophical Society of Liverpool (1881):
I may well maintain that the whole newly-arrayed Anti-Semetic hatred and agitation now going on in Germany can be traced to the following reason: Because the Jews there, as one man, are siding with the party of progress and liberty ; at the same time being dearly attached to the Sovereign and the Royal Family.
And from an undated article in The Jewish Chronicle, quoted in "Letters from Galilee," in Blackwood's Edinburgh Magazine, volume 134 (October 1883):
The return [to Palestine] has formed the aspiration of all the noblest sons of Israel during the Dispersion, and it is not strange that it should still retain its hold on those who inherit their spirit. On the other hand, much is to be said for the opinion that any premature indulgence of this sentiment is likely to be prejudicial in view of anti-Semetic accusations of want of patriotism.
The spelling antisemitic is (not surprisingly) vastly more common in Google Search results, but the earliest match for that spelling is only very slightly earlier than the first instance of antisemetic cited above. From Punch, or the London Charivari (January 29, 1881), page 48:
THE ANTI-SEMITIC MOVEMENT.
Brown. (M.P. for Bloomsbury). "What a shame, this persecution of the Jews in Berlin!"
Sir Gorgius Midas (flaming up). "'Shame?' Serve 'em right, I say! They're all very well so long as they're kep' under, them 'Ebrews are ; but just you let 'em get the upper 'and, that's all!—and their hignorance, their hostentation, and the haaies they give themselves knows no bounds!"
Baron von Meyer (who flatters himself, on the strength of his prsonal appearance that no one can suspect his origin). "Hear! hear! Sir Gorchus! You neffer shboke a druer vort zan zat!"
Joseph Jacobs, writing in Trübner's American, European, & Oriental Literary Record (1883–1885) conveniently lists various contemporaneous discussion on the Jewish Question as being "Anti-Semitic," "Pro-Semitic," or "Conversionist."
Another interesting instance comes from The Catholic World (March 1889):
THE ANTISEMITIC MOVEMENT IN EUROPE
The Antisemitic movement, though hardly ten years old under that name, has thrown a mass of publications on the market among the authors of which we encounter the learned Catholic priest, the professor of theology, the lawyer, the atheist, men of all creeds and no creeds, and hence also a correspondingly great disparity of opinions to the intrinsic merits of the whole question.
English words with '-etic' or '-itic/-ytic' endings
With regard to the second issue, consider the number of (more or less) common -etic words in English:
alphabetic, aesthetic, anesthetic, apologetic, apathetic, arithmetic (as an adjective), ascetic, athletic, bathetic, copacetic, cosmetic, diabetic, diarrhetic, dietetic, diuretic, emetic, empathetic, energetic, esthetic, exegetic, frenetic, genetic, geodetic, hermetic, homiletic, kinetic, magnetic, onomatopoetic, peripatetic, parenthetic, pathetic, poetic, phonetic, prophetic, prosthetic, pyretic, splenetic, sympathetic, synergetic, synthetic, theoretic
versus the number of -itic/-ytic words:
analytic, arthritic, chondritic, critic, diacritic, electrolytic, granitic, hemolytic, hermaphroditic, mephitic, nephritic, paralytic, parasitic, psychoanalytic, sybaritic, syphilitic
All in all, for people learning the word by hearing (not reading) it, I think it is likelier that they will assign it to the -etic class of words than to the -itic class of words, the former being considerably more frequently used and heard in English.
I get the impression that a sizable number of people have been pronouncing (and spelling) antisemitic as if it were actually spelled antisemetic ever since the term came into English.
In one respect, this is rather odd, given that the spelling antisemete is practically unknown to Google Books as an alternative to antisemite. But when people try to pronounce a word they are unfamiliar with, they may adapt it to a pronunciation pattern that they are already familiar with, without making much of an effort to connect it to etymologically related words (such as Semite) that they may have heard used in other contexts.
Regardless of how the pronunciation "antisemetic" first caught on, it is backed to some degree by the spelling antisemetic—which has appeared in published English for more than 130 years.