Should we use A\C in the unused second story of our home?
Almost certainly you should shut off the upstairs AC completely. Ideally you would also seal off the second floor as much as by shutting a door at the top or bottom of the stairs (or installing a curtain across it or something).

The only reasons I can think of to leave it on would be:


If you have any valuable artwork or antique / fragile wood furniture upstairs. Excess heat and humidity could be a problem, especially if the temperatures get really high. 
If you don't have a good thermal separation between the upstairs and downstairs it's possible that by leaving the upstairs unconditioned you'll create uncomfortable hot areas downstairs. E.g. if you have a flight of stairs between the floors, the bottom of the stairs might get warm.
If you have separate AC units for upstairs and downstairs and the downstairs one is very inefficient, you'll be making it do more work.