They do now. There are states issuing driver licenses to illegal immigrants and registering them to vote.
Actually the constitution gives each state the right to determine who is a citizen of that state. This is proved, as elkfersupper says, by the fact that each state can determine who can receive a drivers license. States should(and do) have, if not full authority over their borders with other countries, enough authority to enforce not only fed law but state laws concerning immigration.
They’re doing that, but the feds still have the authority to deport any of those people. I also think other states would have grounds to sue that state if the IDs don’t make it clear the person is a non-citizen.
The USA is being invaded with illegals. Most nations would consider that to be an act of war.