The problem I have with the movie is that it attempts to make this a parallel with the US immigration problem, but fails utterly in understanding why it's a problem in the first place, leading to heavy handed and illogical "messaging".
First of all, the issue with the real life US/mexico border issue isn't an issue of place. In other words, if I'm standing near the border of mexico in the USA, there's very little difference between the actual land I'm standing on. It's all about the same dry desert land, basically. The difference is brought about by who lives where, and how they think and how they run things. USA government is (relatively speaking) uncorrupted and uses its resources well to generate prosperity, whereas Mexico's government is hugely corrupted and just doesn't do a good job providing its people with a chance for prosperity. Whereas in this movie, this concept is either not understood or just glossed over, in that all one has to do is reach a different place (Elysium), and then things are automatically better.
Second, if the problem of this time is really overcrowding, I don't think people would build the great place in space, I think it would be the other way around. People would make the planet great, and then ship off the "excess" people to be crammed into space habitats that are little more than prisons. This would be much cheaper, and facilitate an "out of sight, out of mind" rationale much better.
Third, as someone else said, the real thing the people on earth wanted was access to Elysium's advanced technology, not to have to live there. Why not just steal it? There would have to be a way. After a few decades, someone on Elysium would likely feel sorry for earth's plight and defect with the info, or someone would hack them, or whatever.
Fourth, whatever happened to nuclear weapons? At the very least, someone would have gotten pissed off over Elysium living so high and them living so low and sent a few nukes their way, an orbiting station would be incredibly vulnerable to something like this. Maybe they were equipped with a station sized shield gizmo like Kruger used, but still, you'd think someone would have just decided to destroy the place at some point.
The heavy handed terminology was what did it for me, "Deploy Homeland security!" On a space station it's still called Homeland Security? Give me a break.