On the assumption you were growing outdoors, Basil (Ocimum basilicum) is an annual plant in colder areas - now that it's cold, it's unlikely to grow again. You've not said precisely which USDA zone you're in, but if you expect even colder weather during winter, then its definitely not going to grow again, so you might as well dispose of the woody brown one. You've not said what condition the other basil plant is in, in the other container, but if your temperatures are either 60 deg F or likely to get lower, then you will lose that one as well.

You can grow it from seed indoors, but as it grows, it will require as much sun as you can give it, see here for more information on Basil growing generally https://www.thespruce.com/how-to-grow-basil-plants-1402624

On the assumption you were growing outdoors, Basil (Ocimum basilicum) is only perennial in very warm regions; it's a tender annual plant in colder areas. Now that it's cold, it's unlikely to grow again. You've not said precisely which USDA zone you're in, but if you expect even colder weather during winter, then its definitely not going to grow again, so you might as well dispose of the woody brown one. You've not said what condition the other basil plant is in, in the other container, but if your temperatures are either 60 deg F or likely to get lower, then you will lose that one as well.

You can grow it from seed indoors, but as it grows, it will require as much sun as you can give it, see here for more information on Basil growing generally https://www.thespruce.com/how-to-grow-basil-plants-1402624