I too suffered from this and found it was as much a confidence issue as anything else. I found I was getting a bit sweaty, feeling conscious of it and that was making it worse.
It's a problem called Hyperhidrosis, which effectively means excessive sweating, and it can affect anyone and essentially any body part (although armpits is the most common, called axillary hyperhidrosis).
I went to see the doctor and got a prescription antiperspirant that I can't remember the name of. I had to apply it each night before I went to sleep, one night on, one night off. You need to wash thoroughly under your pits every morning to wash off the residue. After doing this for about a week I had an enormous improvement in the situation, and after two weeks I didn't sweat at all.
I kept up using it until that bottle was gone (I think it lasted about 3 months) and afterwards I was fine for about a year. When I began to notice a little under-arm sweating again I went back to the docs and got another bottle, problem solved.
Nowadays I rarely get a problem, but a visit to the doctors should be your first port of call, and they will likely do the same for you. Only if that doesn't work will more drastic action be considered.
