Check out "damping off" via google. See picture in link.

FataMorgana

Here is a link that might be useful: image

Did you separate the sprouts at all? Basil sprouts are very susceptible to bruised stems if you pinch the stem while separating.

I'll bet a million bucks that "damping off" is the problem, as stated earlier. Many factors contribute to the onset and spread of this fungal disorder, but it is preventable.

At this late date, you might simply want to purchase some plants.