Inspect for Damage
Inspect the stem assembly for damage. Check the rubber washer on the end. If it looks OK, unscrew the packing nut to see whether there is damage to the O-rings inside (Image 1). If the washer is damaged, remove the screw, and replace the old washer with a new one. If an O-ring is bad, cut it off with a utility knife, coat a new one with heat-proof grease, and slide the new O-ring into place on the stem, using a flat-head screwdriver as a guide (Image 2).
An alternative to replacing washers and rings is to replace the entire stem assembly. If you decide to take this course, it's best to replace the stems on both the hot- and cold-water sides. Note that because the hot- and cold-water handles often turn in opposite directions, the hot and cold stem assemblies may be different. Make sure to buy the right parts. Some are color-coded to indicate hot or cold.