Unfortunately, kidnapping is a major concern at borders. If you are traveling with your children they will need some form of identification. If both parents are not traveling with the child you may need a notorized document from the other parent giving permission for you to travel out of the country with them. Likewise, if your children are subject to a custody decree, you must establish your legal right to travel with them. Once again, the best route for any unusual circumstance is a passport.
Children can have their own passports which makes travelling with either parent much easier.
Oh yes, warn your kids that smartass remarks at the border can cause major delays. The immigration agents are generally friendly and helpful, but they don't hesitate to check if they get the slightest hint that a child is at risk. If you are travelling by car, children and parents should be in the same vehicle when crossing the border.