Q.Is basil hardy in S.W. VA?
What temperatures will basil tolerate? I live in South West Va. ; winters can be below freezing.

Basil is a rather short lived annual. You can expect it only to survive the summer and after that it will get woody. It is best to start a new plant for winter to grow indoors. Here are some articles to help you with your basil plant and future basil plants:
https://www.gardeningknowhow.com/edible/herbs/basil/how-to-grow-basil-plants.htm
https://www.gardeningknowhow.com/edible/herbs/basil/growing-basil-seeds.htm
https://www.gardeningknowhow.com/edible/vegetables/cucumber/propagating-basil.htm
https://www.gardeningknowhow.com/edible/herbs/basil/information-on-how-to-grow-basil-indoors.htm