I have restored several outdoor devices, including a garage door opener this way:
Start with hot water, and something gentle, like a toothbrush. The hot water can help solubility compared to alcohol or even cold water. Alcohol (isopropyl, 90%+ ideally) is still good as a final step.
Obviously unplug/remove batteries, and if electronics gets wet, make sure to fully dry it. (better to try to prevent water ingress, partially disassemble, place some paper towels strategically). Also check that the crap didn't get inside and short anything out. The cleaning procedure itself can spread the salty battery goo and corrosion products.
It is even possible to give circuit boards a bath -- but if you do that, dry afterwards with limited heat, <80C. A toaster oven can be nice for this. In principle, shouldn't use for food after such use.
If the contacts do not conduct, sandpaper or some other abrasive may be necessary. But be aware that with the original plating gone, it will corrode again much faster than before. If this is the situation (i.e. you got it working well but had to remove the plating), you can protect it to a degree with some protective gels (typically sold in small packets for use with car batteries). If you can replace the contacts or battery compartment entirely, do that.