Answer: Seedlings may be afflicted by a variety of things, and often it is a combination of factors at work. In my experience the most common reason is that they have been too moist. Here is an overview to help you trouble shoot.
Start out with soil that is just barely moist, like a well wrung out sponge. Plant your seeds and cover them with plastic wrap to maintain humidity, but open it as soon as they start to sprout and put them immediately into bright light. Most gardeners find that a fluorescent shop light with one warm color bulb and one cool color bulb works well enough as long as the plants are kept only an inch or so below the bulbs.
Make sure there is some air circulation as stagnant air can encourage fungal growth. Next, make sure the plants are thinned enough to allow for ample light and air to filter through them. Also, do not over fertilize them. They do not need fertilizer until they have several sets of true leaves.
Avoid overwatering them. The soil should be slightly moist but not sopping wet and you should not wet the foliage if you can help it. You might try watering by dribbling water gently and slowly out of a small pitcher onto the soil rather than spraying them -- wet foliage will also encourage fungal growth.
Finally, make sure from the start that all of your tools and equipment are clean and that you are using a relatively sterile potting mix such as a soilless seed starter of peat, vermiculite and possibly perlite.
Other causes can also be too high or low a temperature. Last of all, you might have luck watering them with chamomile tea when you first see the problem appear -- it might be worth a try even now. Good luck with your tomatoes.
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