It is a basic principle of the enumerated rights of the constitution that they are inalienable (even though THAT particular word comes from a different document.) The right to vote, like all other enumerated rights, exists because the constitution SAYS they are there. That is, the constitution RECOGNIZES the existence of those rights. Not because of a law from Congress or the states. Not from some pulpit-pounder or political podium-pounder. These rights are not bought and sold. They are recognized as the result of citizenship.
A voter ID law makes sense in this limited manner: A polling commissioner is unlikely to know everyone in a particular voting district, so a voter registration ID will help to establish that the person presenting it actually IS a member of "the people" who have a right to vote.
I used the phrase "the people" in quotes because in various Supreme Court cases, it becomes an issue as to how to tell you are a member of the people to whom a particular right applies. "We the people" is not a trivial phrase in the context of USA civil law.
The current furor over illegal immigration is about determining whether someone is a member of "we the people" for determination of all kinds of rights. To the extent that exercising such rights might cost taxpayers money via some subsidy or welfare payment, there is at least some motive to eliminate those who are not members of "we the people." Also, voting when you aren't a citizen is an illegal form of "foreign influence" that we cannot tolerate.
Having said that, it can only be part of our duty to recognize the rights of valid members of "the people" and allow them to exercise their rights - such as voting - without interference. Therefore, other than a voter identity establishment process, you cannot place restrictions on voting.