The capacitors will leak regardless whether you turn it on or not. In fact, they have already leaked, and the fluid is corroding pads and pins and eating through traces. Best to get it recapped before more damage is done.
Is it possible that by preceding the "capacitor plague"[1] they are less vulnerable? I have electronics in the house dating back to the 80s that are doing fine.
The capacitor plague is a whole separate event. As bwldrbst explained, this is a specific problem with the SMT capacitors that Apple used in everything during that time period. The through-hole capacitors used for example in the Mac SE and earlier haven't had the problems with leaking. (Speaking only of the capacitors on the logic board... the larger capacitors on the analog board and power supplies are leaking too.)
A lot of early surface mount capacitors seem prone to leaking corrosive crap onto the boards they're soldered to. This seems to affect machines from the late 80's to the mid 90s. Older through-hole caps can leak too but are more robust or tend to do so through their tops.
Machines made be cheap-ass bastards like Commodore suffer from this a lot.
