Safe to control multiple power sources through a switch through the ground?
You have the grounds of the batteries permanently connected together, so opening the switch won't remove them from the circuit altogether: it will leave a potential difference of 3v between the Arduino's power input at the L293's power input.

That would be exactly equivalent to having a 3v battery with its +ve terminal connected to the Arduino power input, and its -ve connected to the L293's power input, with their grounds connected together - so they're sort of wired in series, with one of them backwards, with a far-too-small supply voltage.

I suspect it won't kill anything but I would imagine it would drain the batteries and probably isn't what you want to do.