How to seal/stabilize crumbling particle board counter top?
The door gasket on your dishwasher may not be sealing - check that for sure.

I would:


Stop using the dishwasher for a week to let the area dry out.
Mask off the formica/laminate in front.
Use a solvent-based polyurethane (exterior grade) on the particle board that is crumbling.  This will soak in and hopefully glue it all back together.
Think about gluing a thin plastic of some sort over the area so that the particleboard isn't exposed to moisture anymore