kitchen sink drip

The faucet "seats" (the part the washers bear down against) are probably corroded or chewed up and are grinding away at the rubber washers each time a faucet is closed or opened.

You can probably smooth out the seats with an inexpensive tool made for the job, like this one:

Jeff

The washer seats are probably all pitted since the 60's and the washers cant seat anymore. If you can completely dismantle the faucets you may be able to regrind the seats smoother, or maybe there is mineral deposit around the seats, but I would just look for a new replica of the old faucet and replace it all.

The washer seats may be replaceable. Dismantle the valve & look inside to see if the seat is a separate piece that screws into the valve body.

--Steve

Steve is right. Most faucets from back then had replaceable seats. Use a flashlight and look at the center of the seat. You're looking for either a four-sided pattern or a hex pattern. If you see that, go to Home Depot and buy a "seat removal wrench". It's a long skinny wrench with flats or hex and it will unscrew the old seat. Then go to a plumbing supply house and buy new seats. They usually run about $4 each. New seats and new washers will solve the problem. Refacing the seat with those cheap tools almost never works over the long term. Just can't get them smooth enough.