Segregation of banks would, of course, depend on the laws of the state you were in, but even outside of the regions that existed under the apartheid regime of Jim Crow, the early 20th century is notable for being the point when African-American banking began to take form. Legal segregation may not have existed, but common beliefs that African-Americans had no assets meant that few financial institutions had any interest in catering to them. Focusing on Chicago, for instance, in 1900, there were only seven African-Americans who identified themselves on the US Census as 'bankers', and no banks which were considered 'black' but over the next few decades that began to change.
The First Binga Bank was established in 1908 by Jesse Binga. He was a wealthy African-American realtor who had the personal financial assets to essentially create a bank out of thin air. Rumor plagued him about the real source of his wealth, as his wife was the sister of a notorious gambler, John V. Johnson, but Binga was a self-made man who had made smart investments in land and property, and settled in Chicago, home to a fast growing black community. His bank wasn't specifically aimed at catering to the black community, and in fact the majority of his customers were white, but it was nevertheless an important step, both in the equal treatment of black customers, and the simple presence of black-owned financial institutions within the black community.
He would see several more branches open up, and although the first black bank in Chicago, he was quickly followed by a several more including R.W. Hunter & Co. Bank and A.W. Woodfolk Bank. These institutions provided openings into finance that had been cut off to black persons, and also brought in stability to those communities. R.W. Hunter, in his promotion of his own company, proudly proclaimed it to be "a substantial enterprise, owned, controlled and actively managed by Colored men and women [that] has developed into an important asset of Chicago."
Back before the FDIC, banking - of any color - wasn't always a guarantee in those days though, although the racial aspect certainly brought in further dangers. The R.W. Hunter Bank had expanded quickly, but then started to suffer and was floated by a loan from the A.W. Woodfolk Bank, which didn't prevent it from going bankrupt. The Woodfolk Bank, in turn, was a target during the 1919 Race Riots in the city, which saw one of its properties firebombed. The resulting run on the bank from frightened investors and depositors shutting it down, although it ended up in receivership, not total closure, under P.W. Chavers, who continued the commitment to "serve African Americans in such a matter as to instill greater confidence in black banks and bankers, produce progress for his race, and accumulate wealth for himself and his family."
Binga's Bank would survive longer, but as with many institutions worldwide, couldn't make it through the 1929 Crash and ensuing depression. The 1920s would also see the biggest advancement, under the leadership of Chavers, with the accrediting of the Douglass National Bank, which was the first African-American bank to be nationally chartered, allowing it to put into circulation US banknotes with the signature of the bank President, Anthony Overton, a source of immense pride within the community at the time. It too, though, would fall victim to the Depression, clawing through for a few more years before closing up shop in 1932.
So anyways, to wrap things up, the early 1900s saw a rise of black banking, not just in Chicago, although that was the focus here. Wealthy African-Americans were able to put up the capital to begin these institutions, and they provided banking colorblind services, as well as employment opportunity in finance, both things which white-owned institutions generally failed to do. The institutions were honest, as any other bank, at the least, but they suffered from the instability of banking prior to the FDIC, and a number of them went under; in some cases as a direct result of racial animus, and in others from the simple mundane sort of failures that can impact any bank. While they existed through, they were an important backbone of the African-American community, not only providing deposit and investment opportunities to black persons of Chicago, but also an important point of community pride and stability.
Sources
Baldwin, Davarian L. Chicago's New Negroes: Modernity, the Great Migration, and Black Urban Life. Chapel Hill: The University of North Carolina Press, 2009.
Reed, Christopher Robert. 2014. Knock at the Door of Opportunity: Black Migration to Chicago, 1900-1919. Southern Illinois University Press.
Weems Jr., Robert E.m "Contested Terrain: P. W. Chavers, Anthony Overton, and the Founding of the Douglass National Bank." In Building the Black Metropolis: African American Entrepreneurship in Chicago, edited by Weems, Robert E., and Jason Chambers. p., 80-98. University of Illinois Press, 2017.