When the States created the federal government via the ratification of the United States Constitution, among the powers donated to the Congress was power to enforce the "Law of Nations."
The "Law of Nations" is a legal term of art. It refers to the customary body of laws governing relations between nations. Within the scope of the Law of Nations is the question of migration from and to other nations. Thus, whatever were the powers of the States prior to the ratification of the Constitution, with the ratification Congress received that power from the States.
So, it seems quite likely, for this reason, and no other, that the federal government -- Congress specifically -- decides the rules for immigration to the United States. Enforcement of such laws as Congress enacts regarding immigration is the duty of the Executive Branch.
This conclusion is supported in the outcome of longstanding Supreme Court cases, although the reasoning of those cases has not looked to the Law of Nations power exclusively, and, unfortunately, has sometimes taken the view that regulating immigration is a power "inherent to our national sovereignty." That troubling formulation fails to accord respect to the role of the States in creating the federal government, and inquiring whether, and how, their original power over migration was transmitted to the federal government.
While the federal government has the power to regulate migration into the United States under the Law of Nations Enforcement Clause, the question of how to manage the current press to admit refugees from the Syrian crisis is not answered in the Constitution. Rather, the specifics of how to treat refugees is governed by federal statutes.
Under the 1980 Refugee Act, the Congress requires consultation with the States as to placement of refugees, but not approval. Settlement of refugees, however, does not mean that the States are obliged to provide material support and benefits to those refugees; in fact, federal funding is the carrot for the stick of placement. So, this administration should be consulting with the States, and this administration should give consideration to whether unwilling states are the best placement for refugees, if being so placed means not having access to federally funded support programs.
Long and short of it, the Governors are right to take a stand, but wrong to think that the Constitution grants them the power to enforce that stand to the complete exclusion of refugees into their States.
One further matter.
Because refugees are lawfully admitted to the United States, they enjoy the equal protection of the laws. That equal protection is guaranteed against federal violation by the implicit requirement of equal protection in the Due Process Clause of the Fifth Amendment, and by the express requirement of the Fourteenth Amendment's Equal Protection Clause.
