In classical Latin, as has been pointed out, ‘c’ always had a ‘k’ sound. Later on, though, that sound softened to a “ch” sound when ‘c’ came before an ‘i’ or an ‘e’, and this was standard by the time Latin was adopted by the western church. It’s also the pronunciation which was inherited by Italian, and is still retained.
The sound softened still further to “s”, and this is reflected in the modern French of <i>principe</i>.
English, of course, absorbed a good deal of French orthographical values through the Norman influence, hence the English pronunciation of “principle” with an s-sound. However, English speakers who learned Latin learned it through the church, and hence used church Latin pronunciation, with a “ch” sound in words like <i>principia</i>. This survived until the nineteenth century, when those rationalising Victorial reformers decided that classical Latin pronunciation was to be preferred.
Using an s-sound would essentially be importing (French-influenced) English orthographic conventions into a Latin word. My guess is that this pronunciation would be more commonly found in the US, where there is a well established tradition of dealing with pronouncing foreign names by applying English orthographical conventions - the long ‘a’ in names like Mulcahy and (before the name, ahem, fell out of fashion, Adolph), the stress on the second syllable in Costello, etc. And this, in turn, I suspect, is the result of the great influx of foreign names, etc, as a result of immigration; they were absorbed into American speech, and the beares of them were absorbed into American culture, by Anglicising the pronounciation in this way.
So, oversimplifying, “PrinSIPia” might be a natural pronunciation for speakers of US English, while speakers of other varieties of English are more likely to acknowledge the Latin-ness of the word by pronouncing it “PrinKIPia” or (now considered old-fashioned) “PrinCHIPia”. But possibly a Latino speaker of US English might favour the “PrinCHIPia” pronunciation, applying Latino rather than Anglo conventions to the pronunciation of the word.