That depends on how long you wish to save the leaves. They would last over a month just by putting them into a ziplock bag and press the air out. If you want to do them longer the freeze them in ice cubes and place them in a ziplock bag.

Like Wally said, dryed basil will lose its aroma(thus flavor) very fast. Some people make pesto out of it and freez it.
I have experimented a different method. Just slightly air dry whole stem( while still soft). Then put them air tight zip bags and keep in freeze. So when defrosted there is some freshness and flavor to it.

If you need to save it for a longer period of time drying it some will kill the enzymes and prevent grown that would change the favor. For using it within a couple months I would not even bother. If you are going for a long period of time then drying would be fine.