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.