Somehow, your switch is delivering + 12 VDC from the WHITE wiring harness wire to the BROWN wiring harness wire at all times, when this connection should be broken when the ignition switch is in the OFF position,
You say the engine will turn over without an ignition key; will it START? If so, the switch denies the short of the exciter coil wire to GROUND, the kill mechanism.
Recommend you Google, "KLR650 Wiring Diagram," study the ignition switch connection matrix on the chart, and check each of the colored wires for their respective connections, depending upon switch position. I imagine you'll discover they're honked up.
How to fix 'em? Messing with the ignition switch to repair it seems like an onerous job; a replacement ignition switch (new or used, with key) looks like the way to go. Then, you'll need your existing fuel tank cap/helmet lock key to go with the new ignition key, or change out all locksets.
All this said, your battery won't be drained rapidly; only the neutral light will draw curren (ON A GENERATION 1). Weird that your headlight and tail light aren't on all the time.