From the Fourteenth Amendment:
Section 2. Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed. But when the right to vote at any election for the choice of electors for President and Vice President of the United States, Representatives in Congress, the Executive and Judicial officers of a State, or the members of the Legislature thereof, is denied to any of the male inhabitants of such State, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime, the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such State.[Emphasis added.]
A state without any voter ID requirement (other than a voter registration) ought to sue the states that have those requirements. I think there's an excellent case that if, for example, 10% of otherwise eligible voters in Pennsylvania don't have a photo ID or a government-issued photo ID, or one with an expiration date, or whatever Pennsylvania law requires, then Pennsylvania should lose 10% of its seats in the House of Representatives.
The threat of having even one congressman kicked out would probably motivate all states to make sure every voter otherwise eligible also has the right ID's. They could include a photo on the voter registration itself, for example.
Apparently Section 2 of the Fourteenth Amendment has rarely been litigated. But even if there is no enabling law for Section 2, and no precedent for one state suing another on the basis of Section 2, the amendment alone spells out the remedy to be applied. About the only thing to argue about is how to determine the fraction of potential voters disenfranchised by the rule. When one state sues another, the Supreme Court has primary jurisdiction.