Basil is a leafy "finishing herb," and is best suited to a last minute addition to a dish. Other "finishing herbs" include parsley, tarragon and coriander. Store basil just like you would flowers: trim the stems, and stand upright in a glass of water.
Chives contain a lot of moisture and can get slimy if just left in a plastic container or bag. Wrap bundles lightly in barely damp paper towel, and store in a resealable freezer bag in the fridge. The same method can be used to store green onions.
Thyme is a hardier herb that can stand up to cooking (like rosemary, oregano and sage). Thyme is great dried too! Tie and hang whole bunches to dry. To use, simply crunch the leaves into stews, soups, or dry rubs.