Unless you are cutting herbs fresh out of your kitchen herb garden, you have to buy them and store them. It doesn’t take many bags of delicate herbs crushed and soggy in the bottom of your produce drawer to realize you need a better solution.
Herbs will stay fresher if their stems are placed in water, much like you do for cut flowers. Place them in a pint-size or quart-size jar and then place that jar in the refrigerator. As you need the herb in your cooking, cut off what you need and place the rest back in the refrigerator.
To avoid hazards of tipping the jar over, choose a jar with a lower center of gravity. Canning jars work well. A pint-sized wide mouthed canning jar will work well if the stems are shorter.
If this method doesn’t work for you, second best is probably dampening a paper towel, wrapping the herbs in it, placing the bundle in a plastic bag, and putting them in your crisper drawer.