The (-) from supply1 goes into ground at 5V, and the (-) from the 12V goes into the same ground.
True
which means I'm putting the 12V into the 5V circuit. And this would be bad?
It would be bad if indeed that's what you were doing but the only real connection between the two is the GND.
I can't describe circuits with words so here's another (and I hope a clearer) pic.
Note that at no time does the 12v (the red wire) get anywhere near a 5v device (unless you screw up while breadboarding :)).
The circruit may need other components depending on what the load is, but that's the general idea. Note also that if you use a relay things will be slightly different.
Rob