"The problem of the color-line" is mentioned in the first sentence in Chapter II of The Souls of Black Folk, "Of the Dawn of Freedom."

DuBois characterizes the problem as "the relation of the darker to the lighter races of men in Asia and Africa, in America and the islands of the sea." DuBois uses the contrast of "darker" and "lighter," not specifically that of "black" and "white," despite his particular consideration of black-white relations in the aftermath of the Civil War. DuBois's choice to make the "problem" less race-specific, as well as his inclusion of a diaspora of "darker" and "lighter races" throughout the world, suggests that he saw the problem within the contexts of both American racism and the racist ideology that had justified colonialism.

DuBois does not actually define "the problem of the color-line." His mention of "darker" and "lighter" races and other continents allows the reader to relate the problem to their specific cultural context. What he does instead is illustrate by example. He writes that "a phase of this problem caused the Civil War" and led to the Reconstruction dilemma of "What shall be done with the Negroes?"

In the United States, the answer to that question came in the form of the Freedman's Bureau:

one of the most singular and interesting of the attempts made by a great nation to grapple with vast problems of race and social condition.

However, the bureau did not last, and Reconstruction ended with the Compromise of 1877. The "problem" between "darker" and "lighter races" in the United States, thus, failed to be addressed properly, leaving it to be a central problem in the new century.

Furthermore, DuBois's use of the word "problem" is complicated by its usage in the first chapter, "Of Our Spiritual Strivings," in which the white man regards the black man and seeks to ask, indirectly, "How does it feel to be a problem?" Here, it seems that the "problem" is the relegation of black people to a second-class status and the connection of black identity to political discord.

This is a quotation from W.E.B. DuBois.  It is best for you to identify the author of a quotation.  A page number does not help without a book title, and sometimes does not help even when we have a book title, since every edition will have different pagination. 

After the Civil War, slaves were supposed to be free. But the Southern states enacted laws, called "Jim Crow" laws, that made it nearly impossible for African-Americans to enjoy the privileges of white Americans.  Similarly, people of Asian race were routinely discriminated against in the United States, as were Native Americans.  As DuBois saw it from his perspective at the beginning of the twentieth century, the biggest problem that the United States would have in the new century would be how to address and resolve the inequalities between white Americans and those of other races and color.  The "color line" that he refers to is the line of race that separated people of color from those who were white, a line that kept people of color from being treated equally in education, in employment, and in public accommodations such as buses, restaurants, water fountains, and hotels. 

As things stood when DuBois made this statement, there was a disconnect between the Constitution, which guaranteed life, liberty, and the pursuit of happiness for all Americans and the abysmal failure of that guarantee for minorities.  Now that we have completed the twentieth century and can look back, it is clear that much of the century was devoted to dealing with this problem. 

I am not sure whether this will be a central problem in this century.  Certainly, there are now laws that make discrimination unlawful, but only time will tell if this is a problem we have really conquered.