You can either tie the stems together and hang the bunches up to dry or put loose branches in a brown paper bag. Or you could make pesto and freeze it.
I always have a tough time drying basil. It always seems to do better if I freeze it in ziploks, it'll be a little limp when you go to use it, but it tastes fresh as ever. Also, I just started putting it in ice cube trays or little containers of water; about 6-8 leaves per container, letting them freeze, then popping the little cubes out and keepingthem in baggies in the freezer.
Let me know how the drying goes; I would much prefer to store it dried myself.
Thanks for this question today. I was just thinking about what I could do with my basil besides make a huge batch of pesto. Although, I've been told that making/freezing pesto is a cinch and one "ice-cube" is enough to cover a pound of pasta. At the price of pine nuts in the grocery store, it better go far! I bought 3oz for $5.50!!! (I told myself it would be a summer treat to go with my new garden basil....)
Last year I ran out of pine nuts and didn't want to pay the $$ for more so I used walnuts. Tasted great -- not really that much different. Since we have a pecan tree I'm thinking about using pecans this year.
I just stuck it in the oven at 350 (I think some people put it on 200) and leave the oven door open slightly ... and let it dry out .. just keep an eye on it and when it seems dried out ...it's done .. it gives a great scent in the house! I do this with oranges and apples too ..in the fall