What are our options for a base under a raised garden bed?
Unless your soil is extremely sandy and wet you won't have to take precautions to prevent the garden beds from sinking. I would imagine the tool shed sinking due to the extra weight and foot traffic it endures regularly.
If you feel it necessary to take steps to keep the raised beds stable you can:
-level the beds by excavating under the high corners. This will keep the corner from moving.
-In addition with the excavating pound wooden stakes made from redwood or cedar at each corner. Leave 6-8 inches above ground and screw that section to the bed.
-This idea will help in 2 ways: attach 1/2 inch hardware cloth to the entire bottom of the bed. This will not only provide alot of surface area (drag/friction), but will prevent gophers and other burrowing critters from entering the bed.