Look, I don't think that voter ID itself is a big deal, but there is significant evidence that the GOP think it gives them a bit of an edge. Consider: vote ID appeared in all the southern states (plus Arizona, I believe, also part of the Act) covered by the Voting Rights Act shortly after the Supremes declared it out of date, a GOP operative in Pennsylvania listed voter ID on video as one of the factors that gave them an advantage, in some jurisdictions student ID is not valid while some gun permits are, laws ending Sunday voting hampered "souls to the polls" voting activities in black churches, and I saw a piece about another GOP activist in the Midwest who quit and went public when he was told that the purpose of the ID laws was to help the GOP. On top of this there was the whole voter fraud snipe hunt, the fantasy of three million illegals voting, Trump's New Hampshire absurdity, reduction of polling places in Latino parts of Phoenix producing huge wait times, and what a judge called the "surgical precision" gerrymandering to weaken the black vote in North Carolina.
It is pretty easy to construct a defensible argument that the GOP is responding to demographic changes they see as unfavorable by trying to limit the franchise. Voter ID is not the end of the world, but it is a solution in search of a problem that - not remarkably - benefits only one party. This is nowhere as bad as what the Democratic Party did in the Old South from post-reconstruction til 1960s when threatened by demographics, but it does resemble Mayor Daley's shenanigans in Chicago back in the day to frustrate reform democrats and republicans. Human nature.