Hi sneutron, I'd check the following possibilities in order:
1) human error
2) need more finger strength
3) vitamin deficiency
4) serious medical condition
(1) is most likely. As you say, if you're finding you have to grip really hard with your left hand, you might consider trying a shallower or narrower neck. Or maybe your action is too high for comfort. Does this happen with "all" basses? Have you taken a trip over to your local music store and checked out various basses, to see how they feel? Have you ever tried a short scale bass? (like, 32 inches)?
(2) is quite possible. If you're not used to having to stretch your fingers a lot, and exert force while they're in unusual positions, it's possible that you may just need to get used to the effort. If you stop playing the bass for a couple of weeks, do your fingers still hurt the first time you play again? If not, how long does it take for symptoms to develop? If this is the pattern, you might try developing finger strength "slowly", like try exercising by squeezing a tennis ball at intervals during the day, or by using one of those grip-thingies that the jocks use.
(3) is quite common, especially in women. Could be a vitamin B or C thing, or it could be calcium or something like that. Have you tried taking vitamins, or do you currently take them? If not, try B and C, and if that doesn't work, try a little calcium ("a little" being the operative word - don't overdo it with the vitamins, your body needs just enough to supplement your food intake).
(4) unfortunately, is a possibility. It could be that you have some kind of pre-arthritic condition, that could even possibly get more serious over time. In that case, playing the bass probably isn't good for you. There's two major kinds of arthritis, one of which is much more serious. Or, there's a lot of things (besides arthritis) that can cause joint pain. Are you currently taking any medications? Lipitor, for instance, can easily cause joint pain. If not, there are other medical conditions that might be worth checking out.
Joint pain is nothing to mess around with. I'm very surprised that a doctor, when confronted by a patient with joint pain, would be so cavalier as to say "take an anti-inflammatory and call me in the morning". If your condition persists, and it doesn't respond to rest, vitamins, or Advil, I would definitely seek another medical opinion. There are doctors who specialize in joint pain, and any of them who are any good will do a complete blood workup as the very first diagnostic procedure, to see if there are any biochemical clues about what might be going on.
Meanwhile, it's probably a good idea to make sure your bass is set up as comfortably as possible, with a straight neck and low action. Oh yeah, and there are also doctors who specialize in music medicine (just like there are those who specialize in sports medicine). So you might want to find someone like that, and take your bass in and play it for a while in front of the doctor. Maybe someone like that can point out something you're doing that's causing you pain.
Good luck, I hope things work out for you.
