The effects could be many:
1) 11% of voting age americans can't meet the I.D. requirement some of these state legislatures are putting through.
a) the good news is that this same group isn't all gung ho to vote anyway.
2) illegal/fraudulent voters (the kind that this law would defeat) aren't enough to swing any elections. It's a miniscule amount of voters and the penalties for it are already steep.
a) It will have a chilling effect on some voters and keep them away. Things like getting a letter from the Secretary of State telling you that you have to provide proof of your right to vote prior to the next election or else you'll be purged. This will turn people away from voting and in tight races, especially local or state, it could be the difference.
3) It could also piss people off that their government is trying to stop them from voting. This motivation will backfire on the GOP and in places like Texas, where the population is shifting rapidly, could have negative repercussions to the GOP for a long long time.