Now for the patience part - you will need to give your basil a good 4 weeks before it will be properly dried. Pull it down before then and it will just be limp.
How do I know if my basil has dried properly? The test to see if you're basil is ready for storage / cooking is to break one of the leaves between your fingers - if it's nice and crunchy and breaks easily it's done. If not give it another week and try again.