The stock starter switch has a yellow/red wire, that is connected through the harness to the starter solenoid. When the button is pressed, it connects the y/r wire to ground. The other small solenoid wire is black, which is "hot" (+12v) when the key is on, so when the y/r is grounded at the switch, current flows through the electro-magnet coil in the solenoid which causes the heavy switch of the solenoid to close, operating the starter. If you disconnect the y/r wire from the solenoid and measure the resistance to ground with an ohmmeter, it should read open until the button is pressed, when it should switch to a low ohms reading. I suspect you will have continuity to ground without the button pressed, and need to start looking at the switch and it's wiring.
Was it like this since you've had the bike, or is this a new development?