If you leave them as-is, they will starve each other. You need to separate them.
I planted about 100 basil plants this spring in a way similar to what you did. I spaced them out a bit initially in the planting flat, but I still had to thin them. (The thinning is easier when they start about 1" apart instead of just being randomly sprinkled throughout the pot.)
What you need to do is called "pricking out". The first two leaves that come out are called the cotyledon -- they're leaves from the seed. Wait until most of the plants have their first pair of "true leaves", and then you should prick them out (transplant them). Get a small trowel -- I like to use an artist's palette knife. One by one, gently separate the seedlings from their neighbors, trying to preserve as much of the roots as you can. Grab them by the leaf! Not the stem, never handle them by the stem. Use the knife to gently tease apart the roots. I use the handle of the knife to poke a small hole in the new pot and gently drop the roots into the hole, then gently tamp around the hole (not too hard). After you're done pricking out, water thoroughly. In addition to helping the plant recover from the root shock, it also helps to settle the soil in the planting holes.
This spring I started about 100 basil seeds in an 11x22" flat, then pricked out after about 3 weeks to wider spacing in more flats, then transplanted outside (using the same method described above) after another 3 weeks or so. We ended up with so much basil by midsummer that it nearly became a household crisis! ;)
The whole process is a little intimidating the first time -- you think you're going to kill them all, but it works out ok. This video is a good look at pricking out basil. He works pretty fast, but you can replay the middle part to get a look at what he does.
If you decide that you don't want to prick out, you must at least thin the plants you have. To thin, you can just selectively cut the unwanted plants off with scissors, leaving the ones you want. Leave at least 6-8" final spacing between plants. It might be wise to do a couple of rounds of thinning -- the first thinning would go to 1-2" spacing, and then to 3-4" and then to the final 6-8" spacing. This gives you a chance to select the strongest plants.