You find that things are getting wet under your kitchen sink. So you need to go looking under the sink to determine where it is coming from.
Put a stopper in the sink drain and run some water in the sink. Place some towels under the sink. If the towels get wet doing this, chances are the leak is either in the faucet or around the kitchen sink strainer and not in the drain piping, unless your stopper is not holding the water back.
If the towels are wet towards the back of the cabinet it is probably the faucet leaking. If they are wet in the center, run your hand or a rag along the drain piping and check for water.
If you determined the issue is the kitchen sink strainer here is a great video showing how to replace it.
If it is not the sink strainer, check out our other articles and videos on where your leak is coming from. http://www.homediyfixes.com/where-is-my-sink-leak/