You shouldn’t store basil cuttings in the fridge. Refrigerators are dark and your basil can’t get any sunlight in there. Basil has been known to wilt in the fridge.
Instead, place any leftover basil in a cup half full of fresh water, and then place that cup by a sunny window. Your basil will last a lot longer, and it may even sprout roots, which means that you can replant it and grow infinite basil