How do I prevent water from dripping onto the surface of the kitchen cabinet exterior beneath the sink?
It seems to be floating off from the front of the sink whenever I clean the dishes, and starts to wear on the wood.
Any ideas?

Thanks.

Aside from covering the wood with something like plastic to protect it from the water as you are doing dishes, I would suggest maybe putting a couple coats of polyurethane on the entire surface of the cabinets under the sink. Then the wood wouldn't be as affected by the water. Polyurethane comes in different sheens such as gloss, semi-gloss, satin and flat. So, you could probably match whatever the sheen is on your cabinets. The only disadvantage would be that the area that would be coated with the polyurethane will probably be a little darker than the rest of the cabinets.

You Should Apply Some 100% Silicone Caulk Between The Sink Edge And The Countertop. Clear Or White Depending On The Color Of You Sink
If It Is Stainless Id Try And Use Clear Underneath The Sink Where The Countertop Was Cut Out, This Will Solve Your Problem