1. Clean and wrap them. Clean herbs as you would clean lettuce leaves: swish them around in a large bowl of cool water to remove dirt and dust from the leaves, lift them out of the water (leaving the dirt behind in the bottom of the bowl), shake off any excess water, dry them in a salad spinner or pat them dry with paper towels, roll them up in clean and dry paper towels, and store them thus wrapped in a plastic bag in the fridge. Herbs treated this way will keep very well for at least a week.
2. Store them like flowers. Put a bunch of herbs in a vase or jar of water as you would flowers. Cover loosely with a plastic bag and put in the fridge. Herbs kept this way will keep several days; the better they are covered with plastic so they don't dry out, the longer they will keep.