First aid, first aid, first aid!
I am just starting out but one of my most helpful things in terms of safety has been my first aid kit. I always carry my personal kit because as a freelance you never know what you are walking into. My college first aid kit was notorious for being poorly supplied (when I first started it was empty!). Larger houses should supply first aid kits, but I worked with a new theatre company (showcase equity) and my first aid kit was our primary one. I always make sure to have it stocked and have used it many times. Even if an actor needs a band-aid for something as simple as a bad paper cut, being prepared and calm with small things establishes trust and helps when bigger issues occur.
Also, I think stage management success is directly correlated with personality. I do very well in emergencies. A stage manager tries to be pro-active (safety checks), but they need to be prepared to deal with crisis and deal with it in a level headed manner.
For example : I once had a stage hand cut herself backstage during house open. I had to take care of her, call an ambulance, comfort her, reassign her tasks etc. before we could start but really the harder job is to keep actors and other technicians calm.
You may have the situation completely under control but remember not everyone you work with might be as calm. Patience, reassurance, and good leadership is key to dealing with a safety situation and also preventing a snowball affect.
Hope this helps!