American Pronunciation
The prefixes anti-, multi- and semi- are pronounced by some Americans with the diphthong long i sound, though not necessarily in every environment. As a woman from Indianapolis writes:
[I]t really can be pronounced either way. It depends on the individual and even sometimes on the particular word multi- is being used with. I, for example, would almost always say "mul-tie" before -talented but I'd say "mul-tee" before -task, -purpose, -lateral, -color...most things, really. There's just the occasional word that, for some reason, "mul-tie" feels most natural with. And the same goes for sem-ee/sem-eye, too.
And actually, when it comes to semi-final, I'm pretty sure I say it both ways.
These alternate pronunciations are subject neither to a regional dialect nor, as you suggest, the sociolect of the Great Unwashed unschooled in the nuances of Latin. Though not necessarily the choice of a majority of American speakers, the long i alternates are not stigmatized, unlike, say, pronouncing Italian with an “eye,” despite two presidents from the South, Carter and G. W. Bush, having pronounced it that way.
The Kenyon-Knott Pronouncing Dictionary of American English, 1949, does not recognize an alternative “eye” pronunciation of multi-, though it does for anti-. The prefix semi- does not have its own entry, but in all words listed with the prefix, only semiofficial has an /ɪ/, not an /i/, because of the following vowel; the rest have a schwa.
While Merriam-Webster — in their own curious take on the IPA — gives the pronunciation /ˌməl-ti/ to the prefix alone, it lists the alternative for multilateral /ˌməl-tē-ˈla-t(ə-)rəl , -ˌtī-/. The prefix semi- has three pronunciations, /ˌse-mē , -ˌmī , -mi/, as does anti-, /ˌan-ˌtī , ˌan-tē also ˌan-ti before consonsants/, both including the “eye.” Semifinal is given all three pronunciations: /ˌse-mē-ˈfī-nᵊl , ˌse-ˌmī- , -mi-/.
Origin
The diphthong in these alternative pronunciations arises in analogy to the way English adopts whole words, rather than prefixes, from Latin. The plurals of second declension nouns like alumni or participles as in corpus delecti are routinely pronounced with a final /aɪ̯/ in all varieties of English, a pronunciation that surely never crossed Cicero’s lips. Given that these three prefixes — especially anti- — are still productive in English, then it’s hardly surprising that some speakers of American English will not parse them as prefixes, but as constituent elements in a compound. This means that someone who would never dream of, say, pronouncing antidote with a diphthong, could easily say anti-abortion with an “eye.”