Wrap a couple of layers of masking tape around any shiny parts of your faucet that the wrench might scratch.
If your problem is a leak around the handle when you turn on the water, follow steps one - three below.
On a faucet with a packing nut, cover it with tape and tighten it gently. Never overtighten this nut.
If the leak persists, remove the nut entirely, replace the packing washer underneath, and reassemble the faucet.
To repair a leaky stem that has an O-ring, follow steps one - three below.
Pinch the O-ring on the stem to get a fingerhold, then pull it off.
Replace with an exact match that has been lubricated with a little grease.
Reassemble the faucet.
Check whether the faucet is dripping hot or cold water. You may only have to fix one handle unit, not both. Then shut off the water supply valves beneath the sink. If they aren't there or if they don't work, turn off the master supply for the whole house. Turn on both faucets to let the water drain out.
If there's a decorative plastic button on top of each handle, just slip the blade of your screwdriver under the edge and twist. It should pop right off. Unscrew the screw underneath and pull or pry off the handle. Sometimes, because of the leak, corrosion makes it stick. Try wiggling it back and forth while prying upwards with the screwdriver blade. But never bang the handle free -- the inner workings might break.
What you see beneath the handle is called the stem. Fit your wrench to the large six-sided nut and loosen it. Then unscrew with your fingers until the whole stem lifts out of the faucet.
On the bottom of the stem there's a screw holding a rubber washer in place. Unscrew it, put in a new washer of the same size and replace the screw.
Rub a little petroleum jelly on the threads of the stem. Then screw the stem back into place, tightening it with your wrench. Put the handle back on and turn on the water supply.