The key to keeping basil fresh and fragrant for days (and even weeks) after purchase or harvest is to not store it in the refrigerator. Basil leaves quickly turn black and slimy and lose their signature spicy sweet flavor when refrigerated. A better way to store them is in a jar of water on your kitchen counter top.
Fill a short, stout vase or jar with 3 or 4 inches of tepid tap water. Stick the stems into the jar of water, making sure to add more water to the jar if the end of each stem is not submerged. Place the jar in a cool place out of direct sunlight. Don't worry if the basil droops at first; it should perk right up after about 12 hours. Change the water in the jar daily. When stored this way, basil will stay fresh for weeks. In fact, if you leave the stems in water, they will eventually root and you can replant them in a pot or out in the garden.