In a word, Very. Because in reality communism did make deep inroads into the African-American community. This Flyer distributed by the Alabama Klan may give some insight into the fear that white southerners face from communists in the south. The Communists were able to make inroads into the Black Community via Sharecroppers Unions and workers collectives offering jobs, loans, and support to the African-American Community when it was being threatened on one side by the Depression and on the other by Jim Crow.

The two best books for you to look at are Robin D.G. Kelly's Hammer and Hoe: Alabama Communists During the Great Depression and Defying Dixie: The Radical Roots of Civil Rights 1919-1950 by Glenda Gilmore which gives a very thorough look into the role of radical groups (Such as the communist party) in the early days of the Civil Rights Movement.

Another book for the relationship between Communism (in the form of the Soviet Union) and the Mainstream Civil Rights Movement of the 1950s-1970s is Black Struggle, Red Scare by Jeff Woods who explains in great detail the supposed relationship between the Soviets and the American Civil Rights Movement.

Lastly, I personally recommend All Gods Dangers ed. Theodore Rosengarten. The book is the oral recollection of Nate Shaw. An Alabama Sharecropper who became deeply involved in the Unionization movement in the 1930s and faced many threats and problems from his work. Its beefy, but is probably one of the best (and few) oral histories from a black voice in that era.