Apply a caulk remover over the visible surface of the old caulk. Caulk removers soften the old caulk but they don't dissolve or remove it on their own. Apply the remover following package instructions and allow it to soak in for five minutes to loosen the caulk.
Insert the blade of a new utility knife between the sink and the counter. Cut the caulk away from the counter, keeping the blade as flush with the counter as possible.
Cut the caulk from the lip of the sink using the utility knife. Pull the strip of caulk out from between the sink and counter.
Scrape any remaining caulk residue with a putty knife. Alternatively, use the hook tool on a plumber's five-in-one tool to remove the remaining caulk.
Clean the surface with a bathroom cleaner that contains bleach but not ammonia. The cleaner removes any dirt or grime that was trapped beneath the caulk, while the bleach in the cleaner kills any mildew spores.