<<<Most of us are keen to keep wildlife out>>>
It does depend on the wildlife - rabbits, foxes and badgers are definitely unwelcome, moles a bit less so. On the other hand, hedgehogs, frogs and toads, hares, slow worms, small rodents such as voles, and just about all birds are very welcome.
I agree that children can climb over mesh fencing, and hop between the strands of tensioned wire, as can your dogs. Sticking barbed wire on the top is not the answer, as they will still go over but shred themselves in the process. As you already have hedges, is it possible to put the fence on the outside of the hedge? What you can't see you are less likely to think of. Our dogs no longer try to get over the fence because it's hidden behind hedgerows.
Using vertical slats would stop them climbing but is prohibitively expensive and would be like living in a prison compound.
Depending on the ages of your children, showing them the boundaries and explaining why they mustn't go beyond them alone - the difference between adventure and danger - might help.