I would want to know if there is a program or initiative in place for employees to provide feedback or concerns with deficient equipment, near misses, or safety reporting. How comfortable are employees talking with no retaliation.
What is the process for correcting problems that occur and is the employee actively included if they brought it up?
I'm still new in my EHS career but this is what I've noticed in my short time and what I would inquire about.
Best of Luck!!

I would ask if they have an established safety committee, common types on injuries, safety training topics, etc. Asking about specific things should give you some sort of an idea. I think having a safety committee would prove that they at least have something established for the employees. But anything specific around a safety program can give you an idea of where things are at.

Well apparently I have been going about this all around. I point blank ask them now what their culture is like from the top down. I also ask what managers are like and what buy in has happened and were they want to be at.

I would ask, "What happens when a review of an event finds that there was a safety policy violations?" You can find out whether they want to learn from the event or punish those involved.

I would ask what their current problem areas are and also what their expectations were to be considered a successful hire after 6 months. This would give insight if they are tracking any safety issues or have given any thought to what direction they are heading with the safety program.