You guys are just assuming a lot. The states don't have a right to sue the government because they don't like it's immigration policy. One thing about immigration, is that it results in more people in the land, and whenever there are more people, there is naturally going to be more total (not necessarily per capita) crime. So obviously, it would result in more prisoners. But that's not the federal governments problem, unless your talking about federal prisoners.
It would make more sense if you were trying to argue that the federal government was being negligent in who it was letting in, and let only people who were damaged sue. But that raises a lot of questions in itself. Certainly, whenever the founders had their immigration policy, and someone came in and committed a crime, the states couldn't sue. And most pointedly, most of the people coming in don't have a criminal record, so if they committed a crime, there was nothing we could go on anyway.
As for the states throwing people in prison, they are wholly responsible for every single person they put in prison. If an illegal goes to Texas and snorts coke and Texas sentences him to life in prison without parole, it's not like the feds are hitting Texas's car, ladyt.