If you want to maximize preserved flavor, try freezing them. Clip the leaves from the stems, rinse, and spread them out on the counter to air dry for 30 minutes. Loosely layer the leaves in a storage bag and freeze.
If you want to preserve flavor but aren’t as concerned about looks, freeze basil in a pureed form. Place fresh cut leaves in a blender or food processor. Process for a few seconds to coarsely chop the leaves. Then add enough water to form a slush and process again. Pour this slush into ice cube trays and freeze. Pop the basil cubes out of the tray and store in a freezer container for later use.