If it's a magnetic drive, it should work if it sounds like it works properly (a stable turning noise). Unless it starts clicking or keep restarting the spin, then its busted.
If it's not a magnetic drive but digital, there should not be any sound and if it makes a sound now, it's a bad sign.
Either way, keeping it plugged in should not be a problem, even if its plugged in constantly to power, unless there's a computer hooked to it keeping it at 100% max capacity, it's unlikely it would die, and even then, the hardware should have been made to even withstand max capacity for a long period of time, or it's just bad hardware. If it experienced an electrical surge whilest plugged in however, it could obviously result in a utter destruction. Usually, there are breakers in your house preventing such surges to reach your appliances though.
Plugging it in to other computers is a good way to find out how the drive is doing, if there is no luck and there is very important data on it, you could always send it off to a data retrieval specialist who have quite a lot of ways of getting a lot of information from broken down harddisks.. But they are costly, so if there's nothing interesting on it, that wouldn't be worth it.