In reply to trolomite:
How sure are you that your pains are related to rock climbing? Have you had a series of definite individual finger injuries? Was a general increase in pains related to a stepping-up of training intensity? Just bear in mind that if you go to see a sports injury specialist of some sort and are labelled as a climber from the start, you may not get anyone thinking wider than that. Primary generalised osteoarthritis and rheumatoid arthritis can start with pains in your finger joints too, and it wouldn't be great to just get physio without actually treating the disease process. I'd advise going to see your GP first, to be honest. If (s)he can satisfy you that it's all climbing-related and nothing else, then you can seek out someone specialised after.
All the best!
Pete