I've always felt string buzz was a bad thing, but ultimately I think it depends on the music. I've always been a stickler for the high/clean action for recording. I have most of my basses set up for no buzz even if I dig waaay in. Plus you get a clearer tone that "speaks" better IMHO with higher action.
But I had a session the other day on a grimey rock thing where I pulled out a Jazz with low, buzzy action(technically speaking, in my book at least, a sub par set-up

, and it just fit the track--I was using a real bright aggressive tone for it--it was a real noisy bratty song overall actually

. It's a mood thing. Woulda sounded like garbage on any number of other types of music. In this case it enhanced the part and we EQ'd it to make it even more nasty.
I do think though you should have one good all purpose bass set up for recording with absolutely no buzz--whether a P or J or whatever is your preferred main axe. You don't wanna be caught in a session trying to adjust a setup while the producer glares at you, or regret your sound later on when you listen back to something where the string buzz is annoying.