I know some bassists (Billy Sheehan, for instance), frequently use their fingernails as part of their technique, and while I used to play a lot with my fingernails I have come to keep mine as short as possible on both hands, especially my right (finger-picking) hand. I like to touch the directly strings with my fingers as it gives me a bit of control on the attack depending on where I pluck the string with my fingertips whereas other things (picks, fingernails) feel like they are in the way. I keep fingernail clippers with me and routinely trim them prior to practices and recordings.
Obviously YMMV, and like 10,000 other things with bass playing there really is no right or wrong way except what works for you and what doesn't.