Take a look at this clause of the US Constitution:
Article I, Section 9
1: 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.
Some may argue that this clause is no longer in effect because its built in date has passed.
I would argue, however, that this clause only prohibits the importation of slaves. It does NOT prohibit the States from controlling their own borders.
If I were a US governor, I would declare a state of emergency and close my borders to all migration except those who could be properly vetted.