I realize the inherent absurdity of arguing about the type of a person none of us know BUT being sentimental does not equal F. In fact, I know multiple ENTPs who are actually quite sentimental themselves. An INFJ would not be quite so off-the-cuff witty. I know type profiles aren't exactly the best sources of information, but even just glancing at a couple of them, I don't see anything which matches with Stern. And Ni? What?! The guy is the epitome of Ne.
Also, I wouldn't go so far as to say it would never happen, but his job itself would be very hard for an I, much less an INFJ. An INFJ would not be comfortable asking the types of questions Stern does, much less asking invasive questions over and over until he gets an answer. I remember a specific example (and there are hundreds) of Stern asking Roger Clemens's trainer probably ten times where he injected Clemens's wife until the guy finally relented and told Stern. An INFJ does not behave like that; that's so classically ENTP I could scream. The whole "come on, come on, tell us" mentality.
On a personal note, Stern reminds me SO much of an ENTP I know, so much down to his being a radio host himself at one point, but I realize that's not helpful to anyone else here.
And I used to listen to the guy on a daily basis, so yes, I do know something about him.