1) Do not allow businesses to hire illegals. Put serious penalties on those businesses who do hire illegals and make the fines based on total revenues so large companies don't just pay the fines as a part of doing business as usual
2) Deport the children with the parents. Sounds harsh, but those children are the responsibility of those parents, not of the US.
Jobs and the opportunities for their children are two major reasons illegals come to the US, from where ever. Right now these are powerful incentives for these illegal immigrants to come to the US. Remove those incentives and there will be a reduction in illegal immigration.
Arguments I expect:
1) Businesses will be hurt without those workers. Maybe on the short run, but in the long run it will be better for all. Will some businesses fail? Sure, but others with better business practices will move in to take their place. Should we stop busting drug dealers because they help their local economies by buying whatever drug dealers buy? No. Illegal activity is illegal activity.
2) Those kids are US citizens! Why are they? Why should they be? They were born here, but their parents aren't legal immigrants. Their parents came here, knowingly, illegally, to have a child here, to take advantage of the systems we have in place. They get caught, they take responsibility for their actions, that includes their children.
Yes that sounds harsh, but these are the hard truths about what needs to be done. Otherwise nothing will change. Both political parties are talking big talk but not doing anything because to do something would take real courage and likely political sacrifice. These career politicians won't risk their sweet jobs to really DO anything. Why do you think Arizona felt like it had to step up and take care of things itself.
While Arizona is definitely stepping into Federal jurisdiction what does anyone expect when the Feds aren't doing enough? If the cops can't/won't take care of the crime, vigilantes or lynch mobs are going to step up. They aren't legal either, but it's a response to the lack of action by the officials. It's the "Do your job or we'll do it for you" mentality. I can't say I blame them one bit.