Help needed... NPN current flow with 0v at Base
Try connecting the base directly to the emitter. 

If that doesn't open the relay, then either you got the pinout wrong or the transistor is faulty.

If it does open the relay, then there's a problem with your ground.

Update: 

As others have pointed, you should put a flyback diode accross the relay. 

This will prevent the relay coil from creating a spike of overvoltage on the collector when the transistor is turned off fast.

If the transistor is faulty, that might be what killed it.