The Atlantic has a very interesting short story on how the rise of the Goldwater movement in 1964 was the final nail in the coffin for Black Republicans (via). Though the African-American community had been moving towards the Democratic Party since the FDR administration, a pace that accelerated when Presidents Kennedy and Johnson enthusiastically supported sweeping civil rights legislation, through the 1960 election there were still plenty of Black GOP voters and prominent Black Republican supporters (including Robinson). Nixon had gotten over 30% of the Black vote in 1960 and Eisenhower received nearly 40% in 1956.
The Goldwater campaign was a choice for the GOP -- to honor the Democratic Party's embrace of civil rights and create a national front on the subject, or to embrace the fury of historically Democratic White southern voters and abandon the legacy of Lincoln outright. They chose the latter. It is often said that Senator Goldwater was not himself personally racist, but that made the choice even more stark -- it was a calculated decision to sacrifice the principle of racial equality (not to mention the equal standing of Black Republicans) for votes in the South. The Goldwater campaign openly appealed to racists in a way that made it impossible for even the most rock-ribbed Black Republicans to continue to support the Party. And compared to those relatively decent numbers in 1956 and 1960, since 1964 no Republican has ever received more than 15% of the Black vote.
There's an important lesson here. The overwhelming dominance the Democratic Party today enjoys among African Americans was not a historical inevitability It wasn't because Black citizens were inherently inhospitable to the Republican Party or its ideas. It was because the Party made a choice to embrace its most extreme elements and, in doing so, made continued Black support untenable. The great migration of Black voters to the Democratic Party was not a blind stampede. It was an advised decision, just as the GOP made its own calculated decision to effectively toss its remaining African-American contingent overboard.