Different herbs demand different storage techniques.
Basil, Parsley, and Cilantro
A fresh bunch of basil can be treated like a bouquet of flowers: Just trim the ends, place in a glass with an inch or so of water, and place on the counter at room temperature. (The leaves will turn black if refrigerated). The basil will remain fresh for anywhere from a few days to a week. You can also try this with similar long-stemmed herbs like parsley and cilantro.
Chives, Thyme, and Rosemary
Other herbs, like chives, thyme, and rosemary, require a slightly different approach. Wrap them loosely in plastic wrap and place them in the warmest part of the refrigerator; one of the compartments in the door works perfectly. Do not wrap the herbs tightly or the trapped moisture may cause them to mold prematurely; many people like to add a crumpled paper towel to the bag as a safeguard. Do not rinse the herbs until just before using.
How to Dry Fresh Herbs
If you have more fresh herbs than you can use, dry them. Place the leaves on a plate (chopped if using basil or parsley; whole if using thyme or rosemary) and set aside in a cool, dry place for several days. Then store them in a resealable container in the refrigerator.
When to Pitch Fresh Herbs
Fresh herbs are no longer fit to use and should be discarded when the leaves turn dark or brittle, or the stems begin to show traces of mold.