Answer: It sounds like the seedlings may have been affected by "damping off" which typically happens when they are kept under conditions that are too dark and/or too moist.
Start out with soil that is just barely moist, like a well wrung out sponge. Plant your seeds and cover them with the plastic wrap to maintain humidity, but open it as soon as they start to sprout and put them immediately into bright light.
Make sure there is also some air circulation where you gorw them on because 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. 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 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. The soil should be kept barely damp, never soaking wet.
Other causes of seedling failure can also be too high or low a temperature or too little light -- many gardeners find it necessary to use supplemental lights for seedlings. A good growing temperature for most plants is about 65 degrees.
Last of all, you might have luck watering them with chamomile tea when you see the problem begin, along with correcting the growing conditions.
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