Why can't I grow basil?

I am trying to grow some basil plants from seed. I used really good seed starting mix and a big planter. I only lightly covered the seeds with dirt and moistened the soil. All of the seeds appeared to sprout. The little plants came out with those 2 big first leaves, but before the true basil leaves came out, the little sprouts start to wither and die. I started them outside in a bright area but without direct sunlight. I haven't moved them into direct sunlight either. It's about 80 degrees every day here in North Florida at this time of year. What am I doing wrong?