Water leaking through vanity, through gap between faucet base and sink
Yes, there is supposed to be a watertight seal. All the valves and spigots are supposed to either have a gasket of some sort or sealed with plumber's putty at the time of installation. They can deteriorate with age. The drain seat should also be sealed. You have two choices: remove the fixture and put on some plumber's putty and reinstall or get some silicone caulk and place a bead of it around the fixture after cleaning the area and letting it dry completely.