Treat fresh herbs like a bouquet of flowers: stems down in a few inches of water. Keep the bouquet loosely tented with a plastic produce bag and store in the refrigerator. This treatment keeps herbs hydrated but not too wet. Many herbs, like parsley, mint, and cilantro, may last up to two weeks this way. Exception: Cut basil is happiest around 55°F, which is colder than room temperature but warmer than the fridge.
Test kitchen experience has taught us that if basil is purchased already refrigerated, it should stay that way. If cut fresh or bought unrefrigerated, keep the basil at room temperature unless it's very hot out. Regardless, cut basil rarely keeps longer than a week.