Our basement ceiling is leaking and we believe its probably from the dishwasher. We are new home owners so this is all foreign to us. What is the best plan of action?

Your ceiling isn't leaking, your pipes are. Tear open the ceiling since it needs to be replaced, can't just let it dry. Once the ceiling is out you will have a better chance of seeing exactly where the water is coming down. It could be a drain or a supply pipe. Does it drip the same amount 24/7? Or does it drip only at certain times?

First, open the cabinet under the sink and look for a water line that goes to the DW. Any leaks? Now look for a plastic line coming from the floor, making a high loop and then going into a drain or into the Garbage disposal. Your locality may require a vacuum breaker in which case there will be a line going to a small device the top of the sink and another line from that back to a drain or disposal. Notice any water there? Finally, under the front of the DW is a small access plate, usually held on by 2 screws, one at each end. Remove the screws, pull the plate out and you should find the waterline and fill solenoid, usually blue plastic. Any drips noticed? If it leaks all the time it is a supply issue. If it leaks only when emptying, it is a drain issue.