Turn off the hot and cold water supplies using the shut off valves under the sink. While not strictly necessary, this prevents accidentally soaking yourself and the kitchen or bathroom.
Inspect the connections of the drain basket to determine if the leak is due to a loose nut, worn washers or dried cracked sealants. The drain basket is the filter at the bottom of the sink, which connects to the pipe below the sink. If you find any structural problems, such as breaks in the washer, rusted nuts, or wear on the basket, then take those parts with you to the hardware store. This will help you purchase the correct sizes when you buy the replacement parts. If none of the parts are damaged, then the gasket or sealant is likely old and damaged.
Disconnect the pipe from the drain basket. Use the slip joint pliers, which adjust to a wide range of sizes.
Remove the locknut on the drain basket with the spud wrench. You may need to tap the wrench with the hammer to break it loose.
Remove all of the old putty with the putty knife. Be careful not to scratch the enamel or metal of the sink bottom with the putty knife. Apply a bead of fresh putty with the caulking gun.
Replace the drain basket in its hole. Place the new rubber gasket up against the bottom of the drain basket and put the metal ring over it. Replace the locknut, tightening it with the spud wrench.
Re-attach the pipe to the bottom of the drain basket using the slip joint pliers. Make sure that all the nuts are tight, but do not over-tighten any of the nuts, which could crack the pipes or the nuts. Allow the new putty to dry.
Once the putty has dried for its recommended time, check for any leaks. If leaks appear at any of the nuts, then tighten them a little more.