It’s best to clean straw hats without the use of liquid. Instead of waiting until your hat is stained to maintain it, try to use a clean, dry cloth to continually wipe away dirt, dust, and debris as you see them. For hat scuffs, you can use a gum eraser, which can be purchased online or at most art supply stores, to gently buff or rub them away.
When you see a sweat or other stain, try dry cleaning the stained area with talcum powder or cornstarch before introducing any liquid to the process. Simply pat the powder on to the sweat stain, wait a few hours to let it soak up the liquid, then wipe it away to see if the stain is gone.