That depends on the chemistry of the electrolyte and the quality of the soldermask, but you'll rarely see more than superficial corrosion.

Batteries are a far bigger problem - most computers have a battery on the motherboard to power the real-time clock, which will do horrendous damage to the PCB if it leaks.

I rarely see the electrolyte get very far from the capacitor, so I’m guessing that it wouldn’t be an issue. Once it dries, it can be hard to remove, but since it rarely makes it far beyond the capacitor itself, I haven’t needed to wipe it up. I’m also a little afraid that using a wet paper towel might just spread the stuff further around and cause a short, but I’m not sure. The biggest issue could be the fact that the capacitor itself probably doesn’t work, which might be problematic if it’s in a PSU or similar.

Since the oldest hardware I have is from the mid-90s, I suspect the problem would be more pronounced in older machines and a re-capping job might be warranted then.