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Immigration has not and never has been an area where states are able to exercise independent authority.
In point of fact, the states exercised the only political authority over immigration into the United States from its founding until the late 19th century.
The US Constitution, as ratified, explicitly forbade the federal government to interfere with state control over immigration for 20 years:
Article I, Section 9: The Migration or Importation of such Persons as any of the States now existing shall think proper to admit, shall not be prohibited by the Congress prior to the Year one thousand eight hundred and eight, but a Tax or duty may be imposed on such Importation, not exceeding ten dollars for each Person.
Additionally, the US Constitution, as ratified, forbade its own amendment to change that for 20 years:
Article V: ... no Amendment which may be made prior to the Year One thousand eight hundred and eight shall in any Manner affect the first and fourth Clauses in the Ninth Section of the first Article ...
That was not accidental. It was debated, and those favoring a federal immigration power lost, primarily because such a power would likely have kept the Constitution from being ratified both in the slave-holding south and industrializing (and therefore immigration-encouraging) Pennsylvania.
After 1808, Congress and the states could have amended the Constitution to create a federal power to regulate immigration, but they never did, so no such power exists:
Amendment X: The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
And that's how it was, and how it was understood until the late 19th century. Congress exercised power over naturalization (how one becomes a citizen), but never over immigration. The closest it came to that was passing a few laws allowing federal port officials to enforce state immigration laws and collected fees or fines to cover the costs of that enforcement.
In 1875, the Supreme Court pulled a federal immigration power out of its ... vivid imagination ... in Chy Lung v. Freeman, and Congress passed the Page Act, which was a very narrow bill forbidding the migration of unwilling Chinese women for sex slavery.
In 1882, Congress passed the Chinese Exclusion Act, which didn't rely so much on the Supreme Court's ... vivid imagination ... of 1875 as it did on renegotiated provisions of the Burlingame Treaty (which, being a treaty, is on par with the US Constitution as "the Supreme Law of the Land").
But, in the main, the states retained near-exclusive control of immigration until the 1890s, and the feds didn't get "comprehensive" on the issue until well into the 20th century.
Is the Arizona law thoroughly anti-freedom, generally stupid, and specifically completely economically insane? Yes.
Is it unconstitutional? In a few ways, also yes.
First, to the extent that it claims to be an enforcement mechanism for federal immigration laws which, per Madison v. Marbury, are void by reason of being repugnant to the Constitution.
Second, in its attempt to conscript employers as unpaid law enforcement agents ("involuntary servitude" prohibited by the 13th Amendment).
But to the extent that it's an immigration regulation law, not only is it constitutional, but it's one of only two kinds of immigration regulations that can be constitutional (the other being immigration regulations created by treaty provision or constitutional amendment).