How do rivers, broadly speaking, function symbolically in The Negro Speaks of Rivers?
A great question.
The rivers work in several ways. Each of these reinforce the other, which is something you would want in a superior literary work.
First, look at the specific rivers named. Hughes names great rivers in Africa and in North America, linking the two continents.
Second, among the rivers named was the Nile, which was the site of a great empire. This links African Americans, who were at that time enduring discrimination in American society, with the founders of an empire. It says, essentially, we weren't always slaves.
Third, rivers symbolize change. Unless they are frozen, rivers are continually flowing and changing, carrying things along. Naming these rivers essentially promises change to those reading.
Fourth, water is often associated with the unconscious, and with the depths of the soul Hughes mentions early in the soul.
So, see rivers, think change, continuity with a great past, and depth of soul.