Water source per se does not cause damping off. But water source can and does affect things. Softened water is often too high in bad salts. From that perspective, tap water is probably better. But and a big BUTT at that - some tap water sucks and it can have oodles of bad chemicals. Check with your local Extension office and/or Master Gardener group.
RO or deionized is better if you have access. Again though - the problem is what is in the water. And regardless - it will have little effect on damping off.
http://www.ipm.ucdavis.edu/PMG...
"Species of the soil organism Pythium are most often responsible for damping-off, but several other pathogens, including species of Rhizoctonia, Fusarium, and Phytophthora, can also cause decay. Decay is most likely to occur when old seeds or seed pieces are planted in cold wet soil, and is further increased by poor soil drainage, the use of green compost, and planting too deeply."
Bottom watering can help.
If you use seed trays with clear plastic tops, remove them for a few hours a day. If significant condensation is forming on them, remove them and let things dry out a bit. As soon as you see germination, remove them. I don't use them anymore.
Water with a hydrogen peroxide mix.
http://www.using-hydrogen-pero...
Media should be sterilized seed starting media. Any type garden soil is verboten for indoor use. And potting soil will be far worse than sterilized seed starting media.
Run fans over your seedlings indoors.
If humidity is too high, run a dehumidifier.
Clean pots with Physan or Consan. If not available use a bleach solution after washing with a soapy water.
While certainly not all daylilies, I start from 500 - several thousand seeds in my basement grow area each year.
