In the Obama administration’s brief to the Supreme Court against Arizona’s Immigration Law, Solicitor General Donald Verrilli invoked the John Jay’s Federalist #3 to suggest that the founders would not want to allow Arizona to enforce immigration law.
"Petitioners assert that Arizona’s status as a border State that is particularly affected by illegal immigration justifies its adoption of its own policy directed to foreign nationals. But the framers recognized that the 'bordering States…will be those who, under the impulse of sudden irritation, and a quick sense of apparent interest or injury,' might take action that undermines relations with other nations, and regarded that possibility as a further reason to vest authority over foreign affairs in the National government."
It is true that Jay was concerned that border states might instigate unnecessary conflicts. However, he was not speaking of them taking vague “action that undermines relations with other nations” as Verilli paraphrased him. Jay was specifically concerned that the border states would “by direct violence, to excite war with these nations.”
Jay wrote Federalist #3 in the days of an open frontier against the territories inhabited by sometimes hostile Native Americans and the great powers of Europe. He made it clear that he was speaking of actual violent conflicts that would lead to war, not internal policies that would cause other governments to criticize us.
Until the Civil War, the states were able to control foreign immigration admissions through their ports. Therefore, when Jay wrote his concerns about border states, he was certainly was not thinking about immigration policy, much less states enforcing existing law and exercising state police powers within their own borders.
Furthermore, the founders would never speak of vesting authority in a “National government.” In Federalist # 39, James Madison specifically he stated that, while some aspects of the Constitution may be national, the “extent of [the government’s power]… is federal, not national.”