Are the anti-immigration laws in Arizona and Alabama constitutionally defensible?
That's a complicated question, and the actual answer might be that it depends what court the question ends up in. The current Supreme Court is closely divided, with a narrow 5-4 lean towards conservative opinions. Chief Justice Roberts tends to side with state power more than his predecessor did when it comes to 10th Amendment issues, but these two laws are more complicated than that.
The key problem I see with the Arizona law in particular is that it actually requires racial profiling. As a white male, it seems unlikely that a police officer would ask for my citizenship papers, and if he/she did, I wouldn't have any to provide. So under the 14th Amendment, each citizen is entitled to equal protection under the law, and to single out a group of people based on their race and identity seems to fundamentally violate that amendment, as citizens who were born in this country but happen to be Latino would no doubt be asked to provide "papers" as well.
Immigration has also largely been a matter of federal policy, and the courts could very well rule that individual states are not allowed to set border or immigration policies. The states will argue that the federal government has failed miserably in that regard, but I don't know that such an argument will be enough to convince the justices.