The Republican party resents accusations of racism and love to turn it back on the Deomcrats as "reverse racism". How can the Republicans be racist? They have a black friend!
Wait, they fired him.
As I said, most Republicans resent charges of racism and don't think of themselves as racist, notwithstanding that they're very, very white and have to search hard for black Republicans1, Latino Republicans, Jewish Republicans... Look, if you have thousands of prominent members and only a few dozen from various minorities, you have a race issue. The Republican party isn't merely overwhelmingly white, it's so white that there has to be an explanation.
The first possible explanation is the null hypothesis: it's an accident of history. The Republican party is white because it happens to be white. People gather with others like them even if they aren't racist. Unless you make an effort to include people unlike yourself in your group (whether it's a circle of friends or your business or school), you'll end up looking like the cast of a show targeted at your demographic.
We know they're not Republicans because half of them are Jewish
If we want to believe that the Republican party is white not simply because of an accident of history, but because they're racially biased, we'll have to actually look at that history. The Republican party was, once upon a time, the party for black America. It was the party of Lincoln. It was the party that freed the slaves, and under whose banner the first black Congressmen and Senators were elected back in the 19th century. In the last 150 years, something has clearly changed. What happened? This guy.
In the background with his back turned, President Nixon. Facing the camera, George HW Bush.
That guy is Harry Dent, the architect of the Southern Strategy, which catpulted Nixon into the White House. After the Civil Rights Act was passed, a large portion of the American population was unhappy, and that portion was the angry, white, racist bloc. The Southern Strategy was, in its simplest form, "We hate niggers, too!", which pulled white Southerners out of the arms of the Democratic party (for whom they had voted for more than a century) and firmly into the Republican fold. In the words of political strategist Kevin Phillips:
From now on, the Republicans are never going to get more than 10 to 20 percent of the Negro vote and they don't need any more than that... but Republicans would be shortsighted if they weakened enforcement of the Voting Rights Act. The more Negroes who register as Democrats in the South, the sooner the Negrophobe whites will quit the Democrats and become Republicans. That's where the votes are. Without that prodding from the blacks, the whites will backslide into their old comfortable arrangement with the local Democrats.
After the angry racist vote won the election for Nixon, it trickled down to the local level, and the Republican party has had a solid hold on the South ever since. It's hard for anyone born after 1980 to imagine, but the South used to be owned, lock stock and barrel, by the Democrats, and for the same reason it's currently owned by the Republicans: white Southerners really, really hate black people. Once they hated blacks and voted Democrat because the Republicans had been the party of Lincoln. Now they hate blacks and vote Republican because... well, they've gotten subtle about it. In the 60s and 70s, a Republican only had to talk about Civil Rights, and that was enough. In the 80s, the code shifted to "welfare queens", and the like. Everyone knew Reagan meant lazy black folks, but Republicans weren't allowed to call them "porch monkeys" any more.
The really interesting thing is that some of their codes and creeds go back to the 1820s. These days the Republicans scream about states rights as a way to fight for their constituents' rights to attack women and minorities in the state legislatures. In the 60s, "states rights" was code for fighting for segregations; "The Fed'ral gummint don't have the right to put censoreds in my school!" But the States Rights go back to the 1830s, as a way for the Southern states to defend slavery. These days school vouchers are mostly touted as a way for locals to get their children's education out of federal hands. Originally, they were touted as a way for white locals to get their children out of federally integrated schools. As a program begun in an effort to continue segregation by any means, school vouchers have had surprising resilience, perhaps because the Republicans have been forced to pretend not to be racist, and have instead spent decades openly courting the white religious vote.
In short, the Republican party has spent more than fifty years as the party of, by, and for white racists. That racism has been forced underground and become less overt as racism has become socially and politically unacceptable. What's the difference, though, between someone who wears racism proudly on his sleeve and one who merely pretends he isn't racist, or even believes it? What's the difference between a platform intended to deprive non-whites of economics, social, and political equality and one that's identical in all respects but is instead proferred as an "austerity" and "small government" platform that merely has, as a side effect, the economic, social, and political segregation of non-whites? Does it really matter that racists have more targets than just uppity negros, that now they also have to worry about "undocumented immigrants"? Where once anti-Catholic hatred was focused on the waves of Irish and Italian immigrants in the early part of the twentieth century, now there's the added bonus of racial hatred as well!
Many Republicans, possibly most Republicans, honestly believe they're not racist, and that their party's policies have no racial bias in intent or effect. However, more than half a century of history simply cannot be denied. Their racism wasn't cured, but forced underground. They didn't stop advocating their racist policies, but stopped calling them racist themselves. Their protests of "reverse racism" are just the old claims that soon the whites would be slaves to the angry negros, but dressed in the language of their opponents. Just as pseudoscience and religious fundamentalism cloaks itself in the language of science, so too do the racists seek to use liberal philosophy as a shield from criticism, but even a brief study of their history, cause, and effect shows their protestations as hollow.
Are Republicans racist? Not all of them. Is the Republican party racist? Without a doubt. Should its policies be ardently opposed? Only if you genuinely want what's best for all Americans, and not only wealthy white America.
1 - Remember, if they're from the nineteenth century, they don't count. Hell, if they're from before the 60s they don't count.