Do you have an identical board that works?, if so compare it with the faulty one.
However, with the components in place it's fairly likely that you will get resistance readings between the rails - through the components down to chassis, then back up through other components to the other supply rail.
Assuming you've cleaned off all the corrosion, then the board shouldn't be leaking - IPA and a stiff brush soon sorts that out.
Aren't batteries a pain for this sort of damage though - as are some types of electrolytic capacitors, which cause similar havoc.