A leak at the base of a bathroom faucet is most likely caused by the wearing down of one of the parts inside the faucet, such as the packing material or rubber washer. Replacing these worn-out parts should solve the leaking issue.
To replace a worn-out rubber washer, turn off the water and remove the handle of the faucet, which is usually accomplished by removing a screw somewhere on the handle. The packing nut and valve stem must be removed from the faucet to replace the rubber washer. Once the rubber washer is in place, waterproof the valve stem, put back the packing material and reattach the faucet.