When you purchase a bouquet of fresh herbs, you can store them much like a bouquet of flowers. Trim the ends of the stems and place the herbs in a glass with about 2 inches of water.
Cover the herbs with a plastic bag to prevent evaporation and place in the refrigerator. Don't place the herbs toward the back of the fridge since this is usually the coldest part.
This method is also useful in keeping asparagus fresh longer.
Do not use this method with basil. Basil should not be refrigerated since it extremely sensitive to cold and will turn black. Just set fresh basil in a vase or glass with some water and leave it out on the counter.
For more tips on keeping produce fresh, check out these posts:
Keep Strawberries Fresh Longer
Keep Asparagus Fresh Longer
How to Store Mushrooms
How to Keep Tomatoes Fresh
Keeping Sweet Corn Sweet
How to Store Avocados
Refrigerate It Right: No More Frozen Produce in the Fridge