It's pronounced "PRIN-sten" (or "PrinceTon") in AmE.
Here are a few "rules" about when the T is pronounced like a soft D in AmE:
When T is between vowels and the preceding vowel is in a syllable that's stressed: city, party, forty, water. (In AmE, they're prounounced SIDee, PARdee, FORdee and WAHder.)
When T comes before a syllabic /l/ sound in words such as petal, metal, and settle. (In AmE: PEDull, MEDull, SEDull.)
In past tense verbs that end in T: educated, excited, defeated (EDyoukayded, exSYded, deFEEded).
I hope my homemade pronunciation guides make some sense!