If the laws were not overly complicated and the IDs were cheap as hell, then I would support them more often. Not because I think they would do anything (voter fraud is practically non-existent), but they would largely be irrelevant. However, in many states it's not so simple. In my state, which has never had even the slightest hint of rumors surrounding illegal voting (we are a Republican supermajority in almost all aspects, by the way), because of economics we have a highly mobile population (property values skyrocketed, forcing families that have been grounded for generations to leave these towns in the last 2 years or less), and we now have a voter ID law. The state promoted it was "easy as pie" (or some other piece of Americana explaining its simplicity) to vote.
The result in the immediately following election? Many multi-generation native-state voters were told to travel hundreds of miles to vote on election day (I'm not exaggerating on the mileage), students were turned away en mass (we have a lot of native state voters, some who recently came from the neighboring state), the Department of Transportation had a server malfunction that mis-identified a large number of voters as living somewhere else, the Secretary of State was warned of these and other issues and publicly said he would not be intervening in the weeks preceding the election, and for the first time in memory the State's political parties had to help direct voters of all ages to the correct destination or give their sincerest apology for not knowing what to do.
The state government's reaction to that confusion? Make it even harder for students to vote and ignore the infrastructure problem that existed.
It was the dumbest thing I have seen in a long time. My state has never had voting problems until state Republicans wanted to follow the national trend. We're a Republican supermajority state that is lily white, less than a few percent are hispanics or Latinos, and the Democratic Party is in such shambles it has a hard time recruiting candidates to lose most of these races, much less win them.
What was the need? What were they trying to prove?