How can I repair kitchen counters that are disintegrating?
If this is the only area that's being damaged then you could try screwing some stainless steel sheeting beneath it to act as a heat and steam shield from the dishwasher.  

You'll need to make sure that the very edge of the steel is not exposed so make sure it doesn't protrude out.  And you may need to put some epoxy or similar right to the edge to stop the steam penetrating the wood at that spot.  Then sand it all round to make it nice and smooth.  It might work, short of replacing the benchtop with something more sturdy.