Voting isn't a Constitutionally guaranteed right in the US, it's a privilege. Although the Supreme Court and various Constitutional Amendments greatly restrict the power of states to disenfranchise their citizens, states do have that power (for instance convicted felons often lose the right to vote.)
Also, voter registration laws have traditionally been an effective way of suppressing African American and immigrant voters[0] (who tend to vote Democrat) so red states tend to vigorously support such laws.
[0]https://time.com/5855885/voter-registration-history-race/
