All you need to do is to pull the DW out and paint the particle board with two coats of primer. Done. If you want to go the extra mile, use a coat of color matching paint on top of the primer so that you never see any of the particle board at all. I'd do this around the sink edge as well. That's a far worse spot for water infiltration than over the DW.

Or you can staple 6 mil poly sheeting to the underside of the counter and dab a bit of caulk over the staples. Some have even made a "cube" of the poly, covering the adjacent wood sides of the cabinets as well.

Or you can buy a condensing dry type of DW that doesn't vent like almost every single stainless interior one on the market. It will be quieter and the tub won't turn orange from tomato sauce as well.