Basil will lose its freshness and shape if you place it in the refrigerator. Cold temperatures wilt plants, basil included. More than that, basil absorbs the smells coming from the surrounding foods. If you know how to treat flowers, you’ll know how to treat basil, too. Do the same things – keep out of the fridge in a container with fresh water. Freeze the herb and use it later, but blanch it first.