Does the Constitution of the United States not contain lines that all men are created equal? If this concept is a precept of the constitution of a country, should the government not ensure that in all states men are created equally and given equal rights under the law?
Quite obviously, no government can actually change what people think, nor can they legislate a person's morality. When the government system, the courts and the laws, combine to act against a race or ethnicity or gender, then historically government has the been the beginnings of change. Forcibly integrating schools was very controversial when the government first ordered it. Now it isn't controversial at all. Women getting the right to vote was controversial at first, now we couldn't imagine women not being able to vote. The Army being integrated by Truman was controversial enough to start a third party bid for his job, now it's a ridiculous notion that we would ever segregate the military.
I am certain the government can not govern racism. However the government has a responsibility to govern the things that racism often leads to. The encroachment on peoples rights based on their race is wrong and people have to be held accountable for those actions.
Racism is a problem that should be handled by government only when it is manifested in discriminatory actions. In other words, the feelings of racism cannot and should not be a governmental issue -- only actions of racial discrimination should be dealt with by the government.
So why should those actions be dealt with by the government? Here's one way to argue this -- they should be dealth with by the government because they are extremely harmful to the victims and they may not be stopped by anything short of government action. For example, racial discrimination in the United States was not unpopular enough among whites for them to force an end to it without government intervention. They did not boycott stores with racist policies in order to force an end to those policies.
In cases like this, government must step in. They must protect unpopular minorities from the effects of racism because no one else will.