Herbs come two categories: hard and soft. Herbs that are soft like like basil, parsley, cilantro, and tarragon are soft, tender at their stems. Herbs that are woody have stiff stems, this include rosemary, oregano, marjoram, and thyme.
The soft ones should be handled like flowers. At the end or the bases of the stems you will want to cut at angle, then place them in cool water that is fresh water, You will want to change the water every day or two if it starts to cloud.
For basil it is best if it is kept at room temperature, unlike other soft herbs, if you have to store it refrigerator it should be loosely covered with a plastic bag.
For your hard herbs you are going to want to keep them by wrapping them first in a damp paper towel, then loosely in plastic wrap before placing them in your crisper.
A final tip: And this is a must…do not wash fresh herbs until they are ready to be used — it will add excess moisture and waste time.