What kind of pills or treatment can somebody get for finger swelling?

Nothing specific: Finger swelling is very generic - it is important to know if it is traumatic, spontaneous and if it affects other parts of your body. The right medication is based on what the problem really is.

Depends on why swell: Swelling may be due to fluid, ie edema and we need to know why there is edema before we can treat it. Sometimes it can be related to arthritis like rheumatoid arthritis or osteoarthritisor it can be related to swelling in the tendons. We call this tendonitis. Each problem needs to evaluated and the underlying cause delineated before therapy, including medications, occupational therapy.