Recipe(tried): Drying Herbs

If you have low humidity in your house, you can try this. Gather a few stems, tie them together and hang them up with the stem end up in a dry place. I hung mine from a ceiling fan (not turned on) it a room I don't use. As I live in Florida, I wouldn't do this unless I am running A/C so much the air is quite dry.

Here is a method I cut from a magazine...used for tarragon.
Gather many sprigs of one kind of herb into a bunch and put it (leafy end first) into a small paper bag. Close edges of the bag around the stems and tie loosely with string. The bag allows the air to circulate while protecting the herbs from light which can dilute their flavor. Store the bag in a well ventilated room until the herbs become dry and flaky. This could take up to two weeks, but check the herbs daily. (I made a little "door" in the bag for this)
As soon as the leaves flake when pinched, they can be removed from stems and crumbled. Store the crushed herbs in labeled, airtight containers until ready to use. (You might want to punch a few hole in the bag, just to make sure the basil gets enough air.)

I have also dried them in the microwave, just put a few leaves between paper towels or napkins and zap them a few seconds, checking to see when they are dry enough. This is my least favorite method, but works when there are only a few sprigs to dry.

My favorite Herb book, published before Microwaves were so popular, also mentions this method. Wash the herbs quickly in cool water, if necessry, only. Be careful not to bruise the leaves. (this would be especially important with basil). Shake off any water, then drain well. Spread the leaves on a rack in the oven at a low temperature. Remember that "low" in most ovens is 140 degrees, so leave the oven door open. Keep a constant watch, for the leaves will dry in just a few minutes. You can use a layer of cheesecloth on the oven rack if you want to.

I think you might want the basil off the stem for this method, I don't think the stems and buds, if any would dry that quickly.

You must be sure the herbs are good and dry before you put them in a bottle, I always inspect my homedried ones carefully before use, as I had some tarragon once I didn't get dry enough and it started smelling funny. I hope this gives you enough info to experiment a bit. It's fun to do, I think.