What do you use to stop water from dripping from the base of the faucet into the cabinet below?

Replace the trim plate gasket with a new seal or add plumbers putty. You may have to remove the faucets from the sink by removing the water pipes and nuts under the sink. Replace the stem seals on the faucet shaft if water is leaking out from under the handle. You may as well install new washers on the valve stems while you are on project.