Summarize:

Trampolines need to air dry to complete the cleaning process. A trampoline should never be washed if the forecast predicts rain at any point during the day as this can inhibit the drying process. Mild cleaners work best for trampolines, as they minimize the risk of damage. Never use a cleaner with bleach or pumice, as this can harm your trampoline. Never rent a pressure washer to clean a trampoline. This can damage the fabric. If you don't have a hose, you can use buckets to wet the trampoline.
Clean your trampoline on a sunny day. Avoid products with bleach or pumice. Use a hose over a power washer.