The prefix comes from a plural form of the Latin quantifier multus 'many'. As with other Latin plurals ending in -ī, the English pronunciations vary.
The Latin /ī/ was pronounced in Middle English as long /i:/ (as in Modern English seen), but the Great Vowel Shift (GVS), which turned Middle into Modern English, moved all the ME long vowels up a step on the vowel chart.
That meant that words like mice and house (/mi:s/ and /hu:s/ in ME), which were already at the top of the chart, couldn't go any further. So, to make room for ME /e:/ and /o:/ (as in seen and soon) at the top, they fell off and became diphthongs. In particular, ME high front /i:/ became ModE /ai/ , while ME high back /u:/ became ModE /au/.
Which is why mice and house are now pronounced /mais/ and /haus/. This means that multi- can be pronounced as in Latin or Middle English as /ˈməlti-/, or in post-GVS fashion as /ˈməltai-/.