There is a sort of continuum of those living in this country. There are citizens, there are legal aliens, and there are illegal aliens. In real life dealing with individuals, it can be tricky fitting each one into the correct category. I also need to factor in that I don't particularly like the idea of citizens having to "have their papers". However, if we could positively identify each legal alien, that would break the continuum and we'd merely need to sort the citizens from the illegal aliens (which has to be easier). So here's what I would do.
Set up a database of legal aliens. Use identity certain technologies like fingerprints, retina scans, DNA, whatever. I'm not particularly disturbed about keeping that kind of information about non-citizens. The first people to go into the database would be the legal aliens. Allow an appropriate amount of time to get them processed. And of course as new legal aliens come in, add them. This includes everyone for any reason - tourist visas and everything. After all, a large percentage of illegal aliens are actually visa overstays.
Then deal with the illegals. Basically offer a visa for everyone who comes forward and applies. I'd come up with some kind of scoring system to identify the most desirable; points for having a job, points for length of time here, deductions for criminal incidents, etc. Except for some who are clearly bad people (an easy example - murders) who simply get deported, everyone's going to get at least a 2-year visa. Best scorers maybe get a 10-year. As they come forward, put them in the legal alien database. Set a deadline (being reasonable about the amount of time required to process all these people). After that, if you're in this county illegally, you go to jail. 1st offense maybe 6 months. Step up from there. If you're in jail, you're not helping your family whether in this country or back home, so you're going to want to avoid that.
Something like the above could clear up the problem with illegal aliens. Then the remaining problem is appropriately setting the limits on legal entry.
