This basin trim piece you are referring to is usually connected to the tail piece flange with a very fine thread. If has been in place for a long time, they can be difficult to remove. You will need to insert something like needle nose pliers in the top and hold the tail piece from the bottom and unscrew it. Clean out all the old plumbers putty and replace it by making a "rope" of plumbers putty in your hand, place it around the top edge of the hole in the basin, then reinstall the trim piece into the tail piece section. Tighten it so it pushes out all the excess putty, and is tight. Clean up the excess, return it to your can. Job done.
Options include looking for discoloration on surrounding areas and even putting a paper towel or Kleenex in suspect areas in order to detect a small leak that may not have been sufficient to create noticeable discoloration. Depending on the volume, you may be able to hear dripping in order to help trace the leak. And you may wish to reconsider whether the initial diagnosis was correct or whether the repair was sufficient.
Here are three causes that weren't obvious to us that you may wish to consider:
(1) Around the tub spigot: We had a situation in which we had a leak from a bathroom on the ceiling of the dining room beneath it. We re-caulked the tub and shower surround, did a small test, didn't see any water leaking, and thought we'd taken care of the problem. The next time someone showered the leak reappeared. We eventually figured out that the problem was around the spigot: Water from the shower was hitting the spigot, and because there was not a good seal around the spigot, water was running behind the spigot, down to the floor, and eventually into the dining room below.
(2) The actual copper supply line connections--if you're fortunate enough to have access to them without tearing out the wall: In another house, one with only one floor, we thought we had a leak in our shower pan, but during a remodeling discovered that the hot supply line had not been soldered correctly and water had been squirting out between the wall and eventually out through the brick in the adjoining exterior wall.
(3) Diversion due to insulation: We had another leak that we thought was due to a problem with the flashing between the roof and adjoining siding. After tearing up the roof and the ceiling of the room being damaged, we discovered that the problem a poorly-sealed nail in the roof itself. However, this location was several feet away from where water damage showed up. In this case, the insulation directly below the small leak was sufficient to keep the water from running straight down. Instead it wound around and into another room.
Best Answer
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.