Likely just needs a seat washer & replacement brass screw (of matching size and thread, brass or stainless). You can do it, but it's old stuff...if something else is wrong, you can get stuck. I would suggest hiring somebody to watch the first repair. Other related problems: damaged "seat" which can be reseated or replaced, and a seat screw that falls apart when you try to remove it. Then you need an extractor to get it out without damage...

If you elect to do it:
Turn off the water. Remove handle (look behind it for something to turn), then escutcheon (it will unscrew), then you will be looking at the valve stem. Next, open the valve (remaining water will pour out) and leave it "open". Next, remove the stem with a stem wrench. Take it out and look at the seat (bottom of it). Replace washer and screw. Reverse order to reassemble. Test.

Note: when they are this old, many plumbers will say they can't get the parts and will recommend replacing the valve. You need an old school plumber that will do a "service call on old tub stems that are dripping."

x2.
start with simple fix & if that doesn't work...
move to next level.
you can buy universal sized washers.

best of luck.

When you have the stem removed, insert your index finger an frrl the seat. if it is rough or with pieces missing, then replace it.
It will pay you to replace the packing as well.

When you reassemble the faucet, make sure the stem is fully retracted.

And I'd you take it apart and can't fix it yourself, just reassemble the old parts, turn the water back on, and call a plumber.

I've found Youtube videos invaluable for this kind of household repair. Do a search and I'll bet you find one -- probably several--on fixing a leaky faucet.

Thank you everyone! The faucet is fixed!

We read over your suggestions and had a non-expert plumber come out to see if it was a simple washer replacement before we called in an expert. Turns out it was. We had him replace the cold water washer too, which was about to break as well. No more wasted water!

Thanks for the update. Glad to hear it worked out.

Thanks homebound! My husband was home and the plumber showed him how to do it and left us an extra washer or two. So if we need to do it again, he knows how and we can go buy a stem wrench.

I really appreciate the detailed answers. Understanding how these systems work helps my brain, and makes me feel confident about tackling problems instead of waiting for them to become a real issue.

Emily, half my sinks have this same issue. You have inspired me to call in another plumber to help (prior to me tackling it myself LOL). I thought I had to get all new valves, faucets, etc, all custom cuz they're antique.....I may love you :p

Most of the older faucets were made to be able to be easily repaired. You can usually fix the with just a new washer and save yourself a lot of money.

UPDATE:

Thanks for all your advice everyone! The new washers worked for about 9 months, then it started to drip again, and at this point is back to a steady small stream. Is that a normal time-frame and I should just change the washers again, or do I need to call a more experienced plumber?

Thanks!

Nope. Now that you learned to remove the stems to get to the washers and know where the seats are.....shut off the water pull the stems and seats (keeping H & C together-I won't bore you with what happens when they are switched LOL) and take the to a REAL plumbing supply store. Once they can identify them you can replace them. If the plumber only replaced the washers, most likely the seats got damaged and damaged seats will eat away at the washers. It's not worth the effort to replace just the washers again, I'd just redo the whole thing for very little $$! Of course, it's now time to buy those wrenches!

Thanks, chucksmom. That's what we did.

No, thank you! It reminds me, time to do mine again in my 70 yr old bathroom. LOL! Glad it worked!

Looks like i have a valve stem that once loosened(by tapping gently on it) should slide out. i’ve already removed the nut over it. Does this sound familiar to anyone?