What's the best treatment for a jammed/sprained finger?

Sprained finger: Splinting & non-steroidal anti-inflammatory meds. For short period of time.

Finger sprain: Swelling from a sprain, bruise or tiny fracture may make the joint stiff and as a result hard to make a full fist or painful making the hand feel weak. Typically it is better to transiently splint the pip and dip figer joints straight when inactive, ice and at the same time work on motion when swelling is down. Nsaids can help as well. Often swelling persists in the small joints for a few months.