Not only does class effect everyone in society, but class is also the future. If, and it's a very big if, individual and institutional racism are somehow lessened, we'll likely find ourselves in a society where there are still large unequal divisions between different classes of people, only the two classes will be made up of people of all colors. Simply eliminating outright barriers to participation doesn't solve society's problems. A great test case is England.
England in the 19th century up to World War II was a very homogenous society. There were Irish, Scot, and Welsh migrants, but that was about it. The big conflicts in American society around race were absent. Yet huge, obscene, inequality existed, and was bolstered by stiff upper lip self help ideology, in the best bourgeois tradition. Self help didn't change anything, even though society was supposed to reward those who worked hard. Instead of self help succeeding, England gave birth to the militant labor movement, and while it didn't press for revolution as strongly as continental socialist movements, it was still as tough as they come. Eventually it formed the Labour party, and after World War II the Labour party took over and created a welfare state in the birthplace of self-help.
England represents the future of America. It is what America will face if outright barriers of racism and ethnic prejudice are dissolved. Not only that, but the same structures leading to the birth of pure class politics manipulate racism and ethnic prejudice to serve them better today.