Someone shared with me how they deal with this in Australia, and I wish they did that here in the US. Employers can't just run background checks and interpret the data any way they want.
You send the job description to the police, and they run the background check and just return the employer a "yes/no" as to whether the person is okay for the job description, based on the specifics of the job and the background.
Edit: Yes, there are concerns about doing this in the US. I thought it was worth sharing anyway, as there are probably changes you could make to get the "good" part of this working here.