A dripping bathroom faucet is almost always a worn cartridge or failed rubber seats. Here's how to fix it yourself.
Turn off both shutoff valves under the sink. Pry off the decorative cap on top of the handle, remove the screw, and lift the handle straight off.
Unscrew the retaining nut and pull the cartridge straight up and out. Note the orientation before removing. Insert the new cartridge the same way. Replace the retaining nut and tighten snugly.
Apply plumber's grease to all O-rings. Reattach the handle and tighten the set screw. Turn the water supply back on slowly and check for drips.