Basil is an annual in most cases. You can pot it and bring it inside for a while, but it's only meant to live it's life cycle out within a year and after a while it will go to seed anyway. There are a couple varietals that are perennial, but typical sweet basil is not.

In addition--the hot temps and direct sun that basil prefers are generally not found indoors.

I find that cuttings of basil root easily when placed in a glass of water. This is how I store my harvested basil for fresh culinary use. I do not bother to bring inside potted plants since in my house they stay alive for only a few weeks.

Harvest it before winter and plant new, strong plants next growing season.

FataMorgana