Yes, wrap everything in red tape and "health and safety", wear a helmet and a high visibility jacket all the time inside the university and even going to bed... That's the answer. Oh, and more stupid courses on how not to break your neck sitting at a desk.
Labs are more dangerous, because they are doing non-standard groundbreaking stuff in the labs, not some conveyor repetitive stuff that people have been doing for 100 of years and every move is known. That's why it's a lab and not a factory- you do risky unproven stuff there. Also, you get young hotshot students/postdocs working in labs, not professionals with experience and a mortgage and a family, so they are more accident prone as well.
I'm not working in a lab, but in my experience accidents happen in following circumstances:
* People are too tired or stressed out.
* People are being rushed too much.
* People don't know what they are doing.
* Well, small number of "Hold my beer and watch this" moments. I guess students are somewhat more prone to those.
So if you want less accidents to happen, make working hours reasonable first (I know post-docs and students in universities work insane hours). And train them better. Of course safety equipment should be available when needed. But more red tape is not the answer, and getting higher-ups involved will wrap everything in so much red tape that getting anything done will require even more hours and frustration, probably leading to more accidents.