I'm not familiar with the G1 specifically, but electrical diag is the same for pretty much any cart.
Check your wiring diagram here
http://www.courtesygolfcars.com/yamaha-schematics
Looks like at the key switch, (with the switch unplugged) you should have continuity between the white, red/white, and blue wires in the reverse position. And the white/green, black, red, and pink wires should also have continuity with each other when in the reverse position.
Then verify the blue wire at the solenoid is getting voltage when in the reverse position, and the black wire has a good ground connection.
If you have voltage coming out of the key switch on the blue wire, but nothing getting to the solenoid then there's a break in the wire somewhere. You could try and chase it down but honestly, I would run a jumper wire directly from the key switch to the solenoid and see if that works. Just drape it over, doesn't have to be fancy or neat, it's just a test. If it works, then it would probably be easier to just run a new wire alongside the factory harness. If that's the case lots of zip ties so it doesn't get caught on something as you're driving.