For me, not hugely often - probably 2-3 times a year. Several of my friends went more frequently - usually for concerts at MSG/etc or pro sporting events, or tapings of Conan or whatever. It did impact my experience, though... if I'd have gone to a school in the middle of nowhere, I'd have lost my mind.
The advantage NYC offers is a host of entertainment options not available at other schools (which only really matters for players with enough disposable income to take advantage of them), plus availability of internships/professional connections (which matters more for players who aren't thinking NFL NFL NFL). If you are a star, and the team is successful, the media attention can be big (e.g. Ray Rice), but the media doesn't usually pay attention to college football.
As for NYC not sprouting out of the sea... you've got it backwards. NYC has always been there, but Rutgers had nothing to sell as a program... only after Rutgers sprouted out of the sea did the proximity to NYC became an add-on selling point.