Remove the old grout with a grout scraper -- a tool with a claw shape that is designed to pull out old grout. Get out as much grout as possible. You won't be able to remove it all.
Brush loose grout away with a thin, nylon brush.
Insert silicone caulk into a caulking gun. Cut off the tip and press a nail into the head to open the nozzle.
Apply a continuous bead of caulk where the sink meets the counter. Insert the caulk as deep into the crack as possible with the nose pointing at a 45 degree angle.
Wet a finger and work the caulk into the crack. Smooth the caulk with the wet finger for an even finish. Rewet the finger as necessary. Allow the caulk to cure according to the package instructions before using the sink. Typically you want the culk to cure for approximately 24 hours.