First things first, you want to try and find cookie recipes that are extremely moist from the beginning. The more moisture a cookie has initially, the longer it will take them to become hard and stale. Exposure to air causes moisture loss through evaporation. It may seem silly to think of evaporation in relation to cookies, but keeping the cookies tightly sealed, away from the open air will help prevent them from drying out.
Instead of placing cookies in a cookie jar, they should be placed in an airtight container that will seal the moisture in. When placing them in the container, you can place wax paper between cookie layers to prevent them from sticking together.
Plastic storage bags are a great option as well. The cookies can be bagged in smaller amounts, which is ideal for keeping cookies moist. By placing the cookies in smaller batches, moisture lost from opening and closing the container will be minimized. Plus, plastic bags often hold in the moisture better then airtight containers since you can squeeze out excess before sealing the bag. If you don’t have plastic bags, plastic cling wrap is another good option. You can prevent moisture loss this way by tightly wrapping up the cookies.
Remember to store the same cookies together. You don’t want to mix cookie types for storage as they will start to take on whichever has a more powerful flavor.