You've got a critter in there. Perhaps some sort of leaf miner or caterpillar. The black stuff is insect frass. If you probe into that little nest, you'll probably see who the culprit is.

I had a similar problem and posted this exact same question back on the GardenWeb boards. The general consensus was that it was a rot. Most that commented said that basil is a very difficult herb to get to survive an extended period of time indoors. The plant I had lasted months outdoors in a pot just like that. Within a month or so of coming indoors to winter my leaved browned and the plant began to die.

Thanks for the reply. They have been inside and haven't been out at all yet. The temp in the room has always been between 65 and 70. The light I'm using is a t5 and it is 6 inches or so above the basil because the tomatoes are a few inches taller. Haven't given them anything but water. I'm really worried they aren't going to make it. Any ideas what I should change or do?

It's hard to tell from the photo, but at first glance it looks like grey mould (Botrytis cinera). The giveaway (and you'll be able to see this better than I can) is bunches of tiny greyish spores. I think I can see some of it right at the top, where the shrivelled leaves are. The narrowing and browning of the crown is a clear indication of some kind of fungal rot. It'll be systemic, which means that any cutting you take from the plant will develop the disease when conditions are favourable (i.e., several hours of wetness and high humidity). And it'll be in the soil, so best not to use that soil again, especially for basil. I would also sterilise the pot, personally. Like Floral says, this particular plant is not salvageable.