Dip the toothbrush in bleach and use it to scrub the base of the faucet where it meets the wall. Get rid of any dirt or debris. Rinse. Let it dry completely.
Set a caulk tube in your caulk gun. Cut off the end of the tube with a utility knife, making a diagonal hole there of about 1/8th inch wide. Slide a nail into the hole to break the seal and release the caulk. Squeeze the trigger until the caulk emerges from the end. Press the release latch on the gun to stop the flow of caulk.
Set the tip of the tube against the base of the one end of the line. Squeeze the trigger again, pushing the caulk out.
Continue squeezing the trigger while running the tip of the gun around the whole base of the faucet in one unbroken line that encircles it entirely.
Lift the gun away and hit the release latch to stop the flow of caulk.
Wet your thumb and run it around the base of the faucet to completely smooth out the caulk and seal it. Let the caulk dry for 24 hours.