DIAGNOSIS You're right to be careful. Harsh chemicals or abrasive pads can damage the plastic interior liner and leave permanent stains or a foul smell that transfers to your drinks.
FIX Here's the shop-teacher method. 1) Unplug the unit and remove all shelves and racks. Wash those separately in warm, soapy water. 2) For the interior, mix a solution of 2 tablespoons baking soda per quart of warm water. 3) Use a soft cloth or sponge dipped in this solution to wipe down all interior surfaces. It's a great deodorizer and cleaner. 4) For stubborn spots, a little mild dish soap on a damp cloth is fine. 5) Wipe everything down with a clean water-dampened cloth to rinse. 6) Leave the door open and let it air dry completely before plugging it back in and reloading.
IF THAT DIDN'T WORK For mildew or a persistent sour smell, after cleaning with the baking soda solution, wipe the interior with a 50/50 mix of white vinegar and water. Let it sit for 10 minutes, then wipe with clean water and dry thoroughly. The vinegar smell will dissipate.
HEADS UP Never use chlorine bleach, ammonia-based cleaners, or scented 'all-purpose' sprays inside a sealed cooler. Those fumes get trapped in the plastic and insulation, and your sodas will taste like a cleaning closet. Stick to baking soda and vinegar.