First thing, sit down with whole family and discuss safety. Use the Eddie the Eagle program from the NRA for the younger ones....If you are not familiar, it has 4 basic rules....
If you see a gun
1. Stop
2. Don't touch
3. Leave the area
4. Tell an adult
Drill that into the kids. Applies at home and anywhere else they may go. Then, make sure the firearm is clear, with no magazine, and allow them to see it, touch it, etc. Use this time to go over the safety rules of handling a firearm.
1. Treat all firearms as if the were loaded.
2. Keep your booger hook off the bang switch (finger off the trigger...keep it fun to keep the kids listening, but emphasize safety)
3. Never point a firearm at something you are not willing to kill or destroy
4. Know your target and what is beyond it
Take the mystery out of it. Anytime my boy asks me about one of our firearms, we get it out, make sure it is clear, let him handle it and I answer his questions.
Drill those into their heads and make sure they understand before you even think about going to the range. Kids that show they understand safety while at the range help us RSO's relax a little...
You might even consider getting a single shot bolt action to start the kids out on, so that you can control them a little. It is easier to teach them to shoot safely when they only have one round. You can put just a single round in the magazine, but that is a pain and the temptation to load up the mag is strong!
Once your wife sees your commitment to safety, and how you teach the kids to be safe, she will probably calm down a little.