Why Does My Faucet Keep Dripping?
A faucet that keeps dripping usually has an internal component that is no longer sealing correctly. The exact part depends on the faucet design.
By Everyday Home Advice Editorial Team

A faucet that keeps dripping usually has an internal component that is no longer sealing correctly. The exact part depends on the faucet design.
Is the water coming from the spout?
A steady drip from the spout points to a different problem than water leaking around the base or handles.
Does the handle fully shut off?
Don't force a handle tighter to stop a drip. Excess force can damage components.
Is the faucet old?
Age can contribute to worn internal parts, but a dripping faucet doesn't automatically require complete replacement.
Repair or replace?
Repair may make sense when the faucet is otherwise in good condition, replacement parts are available, and the finish and body aren't damaged. Replacement may be more reasonable when the faucet has multiple problems, parts are difficult to obtain, corrosion or damage is significant, or you already want a different fixture.
Watch underneath
Occasionally check under the sink to make sure the problem isn't accompanied by a hidden leak below.