Diagnosis
You're right to be careful. The main risk is getting moisture inside the electronics or damaging the fragile ceramic heating element.
How to Fix It
Always unplug it and let it cool completely. For the exterior, just a dry or slightly damp cloth. For dust inside, use a vacuum cleaner hose attachment on the rear intake and front output grilles. For a deeper clean, you can use compressed air to blow dust out from the inside through the same grilles. Do not open the case unless you're technically comfortable.
If That Didn't Work
If there's a stubborn odor or something spilled inside, you'd have to open the case. Remove the screws (often hidden under rubber feet). Once open, you can gently vacuum or use compressed air on the internal components. Never use liquids or cleaners on the circuit board or heating element.
Heads Up
Never, ever immerse any part of the heater in water or spray cleaner directly on it. Getting the ceramic element wet can cause it to crack or shatter when heated, and moisture on the electronics will cause a short.