Apply a chemical caulk remover to soften the hard caulk. Different caulk removers use different formulations, so allow the caulk remover to work for the recommended period of time.
Cut through the softened caulk using a utility knife loaded with a fresh, sharp blade. Multiple layers of caulk may be present, and may require multiple applications of caulk remover. Pull the old caulking material from the joint with a pair of needle-nose pliers.
Remove any remaining bits of caulk from the joint using the pointed end of a painter's five-in-one tool.
Clean the now caulk-free surface to remove mold and mildew stains with a solution of 1/3 cup bleach to 1 gallon of water. Killing any existing mold and mildew spores on the exposed surfaces will help retard the growth of any more mold or mildew.
Rinse the solution off the treated surfaces with clean water. Then dry the surfaces with a clean rag.