I've got some basil plants outside (Spicy Globe Bush), which have gone to seed. I'm wondering if there's any chance the seeds will survive a northern (USA) winter and germinate next year. Is there?
You should harvest the seed and keep in a cold but less freezing place (shelter, car box).
It is in any case good to change place of the basil, and it will also simplify spring work.
So I would harvest now the seeds, keep in cold, and in late winter I would pot them (e.g. in your house). Only later, I would put in full soil. This also helps you to have earlier basil leaves. The plant in any case endure the entire season.
I can't tell you the effect of certain temps on the seeds, but I do know that basil seeds need cold over the winter in order to germinate. I'd take some and follow Giacomo's advice, but I'd also shake some in a pot, on top of the soil, and put a plastic bag over the top to keep out leaves and other seeds till the spring. Then see what they do.