It happens to all of us, we buy very expensive fresh basil from store for one recipe. We need to store the rest of it somehow so it won't go bad. I ran across this tip in the July/August 2006 issue of Cook's Illustrated.
The one method the tester's found worked was wrap all remaining basil in a damp (not very wet) paper towel and store in a zip-top bag, unsealed. Store the bag in the vegetable drawer of your refrigerator. This will keep the basil fresh looking for up to one week.