Bowling Green is not a dangerous place at all. If you have a non-descript car, I would feel comfortable leaving it in any number of places along Scottsville Road: Greenwood Mall lot, Target, etc. Even better would be a 24-hour place, like Wal-Mart, so your car isn't the only one in the lot at the end of the night.

If you're staying at least one night in a hotel, you can ask if it's okay to leave your car in their lot for a few nights. Parking is not at a premium in Bowling Green, so I suspect they'd be okay with it.