State government can't ban another state's citizen. Article IV.2.1 (the "Comity Clause") governs:
The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.
Corfield v. Coryell (1823) first affirmed the right to travel through states under the Comity Clause. Later case law, including the infamous Scott v. Sandford (1857), affirmed that the Comity Clause gives "state citizens, when in other states, the right to travel, the right to sojourn, the right to free speech, the right to assemble, and the right to keep and bear arms."
Most recently, in Zobel v. Williams (1982), the Supreme Court upheld this reading. Sandra Day O'Connor explained:
Article IV's Privileges and Immunities Clause has enjoyed a long association with the rights to travel and migrate interstate. The Clause derives from Art. IV of the Articles of Confederation. The latter expressly recognized a right of "free ingress and regress to and from any other State," in addition to guaranteeing "the free inhabitants of each of these states . . . [the] privileges and immunities of free citizens in the several States." While the Framers of our Constitution omitted the reference to "free ingress and regress," they retained the general guaranty of "privileges and immunities." Charles Pinckney, who drafted the current version of Art. IV, told the Convention that this Article was "formed exactly upon the principles of the 4th article of the present Confederation." Commentators, therefore, have assumed that the Framers omitted the express guaranty merely because it was redundant, not because they wished to excise the right from the Constitution. Early opinions by the Justices of this Court also traced a right to travel or migrate interstate to Art. IV's Privileges and Immunities Clause....Similarly, in Paul v. Virginia, the Court found that one of the "undoubt[ed]" effects of the Clause was to give "the citizens of each State . . . the right of free ingress into other States, and egress from them...."
Wikipedia has an unusually cogent article on the subject of freedom of movement.