Fresh herbs are so tasty in foods that I love having them on hand. One of my favorite ways to keep them ready for use is either to grow them right outside my kitchen to pick as I need them, or cut stems to place in a glass or vase of fresh water. Basil and parsley keep really well this way for several days to a week. Snip as much off as you need and add to your dish. Just make sure you change the water every day.
Herbs will keep even longer if you put them in water, cover with plastic wrap and place the entire container inside a plastic bag, seal and store in the refrigerator. You also can just put herbs inside a plastic bag and store in the refrigerator. They won't be as green and may wilt a bit, but as long as they don't turn black or moldy, the herbs are still good.
Keep the leaves dry. Make sure you wash just before using them. If you store the leaves while wet, they will not last very long.
Another way to keep herbs is to dry them. You can hang bunches of them upside down until dried and store in in plastic bags in the freezer. I plan to dry some herbs soon because even though most herbs will come back each year, they will die down after the first frost, then flourish in the spring.