I have had excellent luck freezing the leaves. Take them off the stem and pop them in a ziplock,then freeze. When you need some for cooking, you can easily break off what you need and add to your dish.
I have also read that many folks recommend making basil ice cubes - freeze leaves in ice cube trays (with water of course) then add cubes to your cooking. This isn't very practical for me because I save a lot of basil, but I have had friends try it.
You can also make nice herb butter with basil and anything else that sounds good. Mince herbs and mix with softened butter - I usually make a pound at a time. put butter mixture on wax paper and roll into a tube - it is similar to having a roll of cookie dough. Place the wax paper rolls inside a ziplock and freeze. You can thaw them out completely, or cut off slices to cook with or add to toast etc.
You don't have to freeze these, they stay in the fridge for weeks, but it is easiest for me to make a large batch, freeze some, and then have them for company or to give as hostess gifts with some fresh baked bread...and of course, some nice red wine
Cyn