The first thing I would do is give it a very thourough cleaning. Some dried up old grease in the sear notch on the hammer can cause this problem.

If that doesn't solve the problem, it's best to find a gunsmith who is both very good and highly recomended for working on old Smith's. Note, now that semi autos are the most popular handgun, it may not be easy to find a local gunsmith who is truly qualified to work on an older revolver, make sure you obtain and check references before handing your gun over to someone you found in the yellow pages. Best case re-dressing the sear on the trigger will solve the problem. Worst case, you looking at having a new hammer and possibly a trigger fitted to your gun.

As for doing it yourself, re-dressing the sear on the trigger isn't very difficult. However, you'll need a microscope to inspect the sear notch on the hammer and I don't know of any way to repair a hammer with a damaged sear notch. Which means fitting new parts to your gun if there is too much damage, in which case I'd have a Pro do the work.

Bottomline, even if it's a potentially easy fix, due to it being a Safety issue, it's better to spend the money and let a real pro do the work.