I think the only problem with undocumented immigrants is that they are undocumented. If we make it easier to become a legitimate US citizen (instead of having to wait for 10+ years), the economy would grow and many social problems would go away. Let me elaborate:
Firstly, illegal aliens usually work for less than a regular US citizen, because their contracts (many times non-existent) are not regulated by the government. And, at any time, employers can just blackmail illegal aliens into a lower wage due to fear of deportation. On the other hand, if we provide a pathway to citizenship to these immigrants, they can sooner be out of that situation and become a productive member of society.
Secondly, we can grow the economy by making these immigrants lawful citizens of the US. Immigrants will become small business owners and create jobs. I think this is pretty obvious and needs no further explanation. Just like regular people with a standard education and standard income can create jobs, after a couple of years (which may vary depending on government action like education subsidies), immigrants will be able to do the same.
Thirdly, providing a pathway to citizenship would increase tax revenue. Many illegal immigrants already pay taxes. This would not be countered by rising expenditures of welfare programs.
Finally, I want to make the moral argument. Illegal aliens in the US should be able to get the rights of a US citizen because they are already contributing to the greater good, both by providing cheap labor through wage slavery (unfortunately), and by paying taxes. Providing a pathway to citizenship is a win-win situation where everyone will profit.