Generally speaking for corrosion I try to use something acidic at least to start. Since even with the damage, I assume there is still some of the bits of corrosion left in the crevices. Most of these sorts of batteries are still using alkaline components so you'll want to neutralize whatever is there with an acid since this sort of leakage/corrosion is an irritant at the minimum. It looks like the cover is metal of some kind and/or plastic? So vinegar shouldn't harm that.
You can use Cotton swabs to actually clean up the corrosion. Or an old tooth brush if you need some abrasion. Isopropyl alcohol will probably help clean everything up too. Obviously the more you can remove the better, since anything left can interfere with electrical contact.