How should I deal with an unused water line that's dripping?
That looks like a 1/4" copper tube with a kink in it. The kink may or may not be what's been holding the water back all this time.

A combination of fittings could be found to cap that tube. A 1/4" compression to male pipe thread adapter followed by a female threaded cap could get the job done, for example. It would be recommended to cut the tube upstream of the kink, though, because the tube is weakened at the kink and is more susceptible to failing.

The better approach is as Alaska Man wrote in a comment: there should be a valve at the other end of this tube. Likely locations for the valve include under the kitchen sink or somewhere near the water heater. The valve may need to be closed more tightly, repaired, or removed. If you can located the valve, or at least candidate(s) that might be the valve, post pictures of it/them and we may be able to advise further.