Neem will surely kill them. What you must do is stop watering so frequently and increase the air circulation (place a fan in the same room but not directly at seedlings). If you have a clear cover on the seedlings, remove it. Check to see if the seedlings need to be transplanted or thinned. If the problem still persists (though I doubt it will) mix 1 part hydrogen peroxide to 10 parts water in a spray bottle and lightly mist the seedlings. Don't overdo it. Hope this helps.

i second the fan, although i place my fan directly in front of the seedlings. the breeze helps them develop strong stems, and prepares them for life outside.

I have been able to halt some mold by spraying with a mild solution of water with about 33% hydrogen peroxide. There were other seeds that I couldn't stop the mold after spraying them so I removed those to save others. Another trick is to place a very thin layer of sand over the top of your planting medium. For some reason, this has worked wonders for me. Fans are a really great idea if you are starting seed inside.

Thank you for all the great advice! I bought a small fan from Home Depot and on the way out, picked up a few more seed packets. : ) Starting seeds is really fun!

Sounds like you've already gotten loads of good, veteran advice, but I'll add my vote anyway for improving air circulation. Using a gentle fan while the seedlings are starting out also seems to strengthen the stems a bit and make the plants sturdier before they head outdoors........Have fun!

I dip all but the finest seeds in strong tea, and then plant in vermiculite. Haven't had molds, mildew, or damping off in years.

Once sprouted, I ditto the fan - if you can find one, the oscillating type seems to work very well.