Package and ship cold foods inside a polystyrene or insulated cooler to prevent and contain melting. Put ice or dry ice inside sealed containers surrounding the cold food, and seal everything so it remains waterproof against the outside of the shipping container. Other techniques include using cold gel packs against the food items or specially made one-use packages. Try to use the quickest shipping time possible to avoid spoilage.
One simple packaging method includes placing the cold food and dry ice into a cooler, and then putting that cooler into a cardboard box thick enough to handle the weight of the cooler. Another way to insulate the cold food from the outside includes lining a cardboard box with rectangles of polystyrene. Place the cold food in the box surrounded by cold packs.
Cold packs must be sealed so they don't leak. One way to do this involves plastic zipper bags or plastic wrap. Create one layer of coolant around the inside of the cooler, put the food in the container, and surround the cold food with more cold packs. Make sure to place a single layer of cardboard between the food and the cold packs. Seal the cooler with tape to help preserve the coolant during the trip.
If you have a cooler packed inside a larger box, use packing peanuts to keep the cooler from sliding around inside the box. Dry ice is a hazardous material, so individuals must inform their shipping company of that fact before shipping.