W/o a pic showing how the handles are mounted, but guessing it's nothing modern nor a cartridge type: take it apart and replace the washer or washer-equivalent. In fact, do both the cold and hot valves. Shut off the water, open the valves in the highest faucet, open the leaky valve and its neighbor, unthread the nut behind the valve handle and remove the bonnet. You may find that the nut will be totally loose and the bonnet won't come out; just turn the handle and it will; the nut has finer threads than the valve stem. March the whole bonnet down to your favorite supply house and buy the replacement washer or washer-equivalent.
My Taiwanese-made fixture had a rubber cup that popped over the end of the valve stem and is not available as a direct replacement. However, it was possible to buy a very similar cup and trim the length of it shorter. Works fine. Interestingly, my bonnets did not have a gasket and they did not have an additional packing nut like old American valves do (normally, American valve stems come out with removing the packing nut, and you don't have to remove the bonnet). The valve stem is made watertight with an O ring, which you should replace, too. Then thumb a smidge of pipe dope on the underside of the bonnet before you reassemble it.
Piece of cake, once you realize you cannot get an exact replacement washer and modifying what you can get works fine.
