To keep cookie sheets from rusting make sure they are dried after washing and store them in a dry environment.
After washing the cookie sheets, dry the cookie sheets completely using a paper towel or cloth and be sure to dry the edges and underside of the rim of the cookie sheet where the water can collect.
You can also apply a thin layer of cooking oil to the surface of the cookie sheet which prevents the cookie sheet from rusting.
You can also apply some rust proof primer to the surface of the cookie sheet that is made for heat.
And store the cookie sheets in a cool and dry place and even a dark place to minimize the exposure to moisture.
Avoid storing the cookie sheets in damp or humid areas.
Also clean the cookie sheets promptly after use to prevent any food residue from creating a breeding ground for rust to develop on the cookie sheet.
You can also line the cookie sheet with a silicone mat or parchment paper to protect the cookie sheets from grease, stains and food residue which can also contribute to rust.
After washing the cookie sheets you can also place them back in the oven to dry them out completely as the heat from the oven will help evaporate the water and dry the cookie sheet off.