Before you start work, turn off the water so that you can work on the faucet without running water getting in the way. Close the drain in the basin so that you do not drop any small parts down the drain.
Remove the faucet's decorative cap or handle to access the parts inside. The caps for most types of Kohler faucets can be simply removed by pulling upwards. For ball-type Kohler faucets, you may need to use adjustable pliers to remove the rotating ball.
The most likely cause of a leak in a Kohler faucet is an O-ring that has worn out. Remove the O-ring from the faucet and replace it with one of the same size, coated in plumber's grease to help it slide more easily over the internal part of the faucet.
Reassemble the faucet and turn the water back on, then turn it on and off again to test it. The new O-ring should help you to fix the leak successfully.