Burpees sell a seed starting tray that has a wicking mat under the tray. Seemed to work really well.
http://www.burpee.com/seed-sta... I've used mine two years in a row now.
I hardly ever have problems with damping off, even here with high humidity! I always water with chlorinated tap water, which I am sure helps. I used to be religious about watering my orchids and other sensitive plants with rainwater, but the orchids suffered plagues of diseases of a fungal nature so I stopped. Improved things immensely.
I'm sure that we just have every fungus known to man happily floating around in the air here. When it rains the spores are washed down, and of course I collect them in my rain barrels and cisterns. Now I just use that water for my veggies (after they are past the transplant stage, and for my Earth Boxes (bottom watered).