I put some auto/marine sealant made by DAP on the underside of my countertop. Just put some on your finger and coat all of the wood under the counter. It doesn't hurt to go out about 5-6 inches on either side of the dishwasher. http://www.dap.com/media/product_photos/AutoMarine.aspx

Thank you, Glass. That sounds easy, which is what we need right now. We'll likely end up doing that, as well as contacting the manufacturer. I just don't remember this much steam and moisture coming out of the DW in the old kitchen.

I used some of that Redguard rubbery paint on vapor membrane that you use underneath shower flooring and walls. It's pretty thick and substantial. We have butcher block counters, and the steam from the DW hasn't caused a problem yet.

We had a bucket of it left over from a project:
http://www.custombuildingproducts.com/ProductCatalog/SurfacePrep/WaterproofingAntiFractureMembranes/RedGard.aspx?user=diy&lang=en

Salesperson showed us two brands that had the steam vents on top the door. They pumped their steam right into the countertop. If folks have a laminate top, they become junk in short order. ( I cannot remember those two brands) Other brands had the hot air/steam vents on the front of the door and there were no problems.

I think the problem is fixed. We cleaned the gasket, moved the DW forward a bit, and the plumber gouped on some silicone on the underside of the counter. I think now that the door also wasn't latching properly (perhaps aided by sneaky little hands). All steam is now properly coming from the vent under the handle. Thank you all for your help!