I think you mean the digital pin to the base? (Edit: I see you fixed that already. It's good practice when you edit, to show what you changed... else others will think I'm imagining things....)
The npn requires the load on the high side, so from the top down, you have:
Power +ve
Load
Collector
Emitter
Ground (same as power -ve)
Then on the "side" so to speak you have:
Controller digital pin to base (through a resistor)
Controller ground to same ground as the transistor, which is the same as the Emitter.
But I don't really understand your question.....
The grounds need to be common so that all voltage measurements have the same reference. If the controller ground isn't the same as the equipment ground, then the transistor has no 0 for the controller io signal, and it's a bit like imagining one hand clapping.
Edit: I guess it's easiest to say that the Arduino ground, the power supply -ve, and the emitter are the same point.