A couple things I would think: First, basil is an annual, and although you can extend it's life by pinching the flowers and leaves regularly, it's not meant to live for more than one season really. It may just be winding down. But...Second--41F is too cold for basil. It's not enough to kill it outright like a freeze would, but it does tend to send it into a dormant type state. I would try to keep babying it a bit, and it may come back or may not. The small leaves may be either the end of life cycle, an I got too cold thing, or something else entirely. Basil also does require a lot of light, so placing it out of direct sunlight will not be good. Hopefully, someone else can address the wrapping in plastic part--I would think it could smother the plant and lead to pest/mold issues, but maybe for just the night it's fine?

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.

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.