The Republicans could have moderated their anti-minority policies, and there were some national party leaders that wanted to do that. But that didn't sit well with their racist teabagger base, and the congressional Republicans have decided to protect themselves (in their party primaries) by going along with the wishes of their base. So, in keeping the anti-minority policies, the GOP had to come up with a different tactic to blunt the demographic reality.
They decided that democracy wasn't as important as hanging on to power, so they began to pass laws to prevent as many minorities as possible from being able to vote. The best known of these voter suppression measures are the Voter ID laws. The GOP knows that this will affect the minority voters more than other groups, since they are the most likely to be without a picture ID.
But that is not all they did to try and suppress voting. Another measure that has been passed in many red states are regulations that make it harder for groups to go out and register voters. This has taken many forms, from intrusive background checks to state registration requirements (and even some silly rules like no one but the person being registered can drop the registration into a mailbox).
The charts above, made from a recently released Gallup Poll (conducted between July 1st and September 26th of 2013 and surveying 43,825 adults, with a 1 point margin of error), show why the GOP thinks voter suppression efforts are needed. While most Whites (85%) and Blacks (81%) are registered to vote, the same cannot be said of Asians and Hispanics. Only 60% of Asians and 51% of Hispanics are registered to vote.
The Republicans are already losing the demographic battle in this country. If Asians and Hispanics were to register in the same percentages as Whites and Blacks, over 80%, it would be an electoral disaster for the Republicans. They would be either forced to moderate their policies, or fade into the pages of history (as the Whigs did).