They used it most prominently for propaganda purposes. Domestically they would highlight the discrimination of African-Americans, particularly in the South, as evidence of the oppressive and unfair nature of the American system. Soviet new reels would emphasize the obvious hypocrisy of U.S. race relations and paint the U.S. as a cruel country ruled by a small circle of wealthy elites.

The Soviets made frequent reference to discrimination in international forums particularly to rebut US criticism of its own abuses. When Secretary of State Byrnes attempted to castigate the Soviets in the UN for failing to extend the vote to people in the Balkans, the Soviet responded that blacks in Byrne's home state were also often denied the ability to vote. The issue also arose during negotiations over the UN Convention on Human Rights as the Soviets point to discrimination in the US as equivalent to the sort of suppression that occurred within the USSR.

Race became a major issue for the U.S. in the wake of decolonization as the presence of non-white ambassadors created problems in a still segregated DC. Many hotels and restaurants made the diplomats "honorary whites" but many still experienced both informal and formal institutional racism. The Soviet Union used this to enhance their own status with the third world and attempted to cast the US as an unfit partner for many of these emerging states. Racial issues in the US also caused many of the Bandung nations to criticize the US and helped provide a moral argument for their non-alignment or even in some cases for accepting Soviet aid.

The long US support for South Africa and its apartheid regime also provided propaganda for the Soviet Union, particularly as it supported Angola and Somalia in the 1970s. They could accurately accuse the US of favoring a racist regime out of expediency further showing the hypocrisy of the US system.

Thomas Borstelman's Cold War and the Color Line is a fantastic and accessible look at the issue.

TLDR: Racial tensions provided the Soviets the ability to highlight the discrepancy between US rhetoric and actions, helping further Soviet agendas both in the domestic and foreign spheres.

Anti-racism and the evils of the U.S. system of segregation was a substantial aspect of soviet propaganda, particularly in the decade leading up to WWII. They wanted to portray the Soviet Union as a country where racial discrimination did not occur, and contrast themselves with the capitalist system that led to Jim Crow laws and segregation. They actually didn't need to make anything up for their propaganda, the U.S. was so awful in that respect.

I suggest you read the book Opposing Jim Crow (or at least the excerpt here) for more details.

Maybe "take advantage of" goes backwards in my example, but a number of black Americans through the 20th century went to the USSR, either for training in Communist politics (which they then exported to the US' Communist party) or just to escape American racism. They seemed to really like it! In fact, in the 50s and 60s, the US started monitoring and restricting the movements of prominent black leftists (who happened often to be entertainers), like Paul Robeson and Marian Anderson.

Sources: Defying Dixie, by Glenda Gilmore has the early century stuff, while Mary Dudziak's Cold War Civil Rights discusses the midcentury stuff.

As an aside, this is actually one of the overarching plot points of FX Soviet era spy show The Americans