The answer is that it depends. As far as the apartheid regime of the Jim Crow states was concerned, it was much more important to focus on the racial separation than issues of gender. For white society, the only reason a black person would be expected to be in a bathroom facility labeled as "White" was, of course, to clean it. In an interview about her experiences growing up in the segregated South, one woman poignantly illustrates the issue:

You just didn’t have to do anything to them for them to not like you. They just didn’t like you because you was Black. And its kind of sad that we had went to a restaurant that we couldn’t go in the front door, we had to go in the back door, and sat there where they washed the dishes at […] So that and then the bathrooms, you could go to, you know, get up and go to a bathroom like the other peoples went, other White peoples went, you had to, you know, go around and go outside, and go to the bathroom, you could not you know, use their bathroom.

What was important was that there was a "Colored" Bathroom and a "Whites Only" Bathroom. Although, infamously, the legal justification for segregation was "Separate but Equal", when put into practice it was generally not even close. While large public buildings might provide gender-separated bathrooms for both races, this required the necessary space and planning, not only to provide for the facility itself, but also for the way to access it, as it was often not even considered proper for them to simply be next to each other. It wasn't even a matter of black and white facilities, but also black and white spaces.

The general result is that it was not uncommon to see "White Men", "White Women" and simply one, general "Colored" facility. The likelihood of gender-separated restrooms also depended on the state. Whether followed or not, the laws of states in the upper South such as Kentucky legally required it by statute, while states in the Deep South like Alabama had not even that meager a protection, although in some locales, at least, it was sometimes allowable for black women to make use of White facilities if there was no alternative, even though it was a concession never made for men.

Discussing this separation of course assumes they even had one at all, as smaller establishments wouldn't even have facilities for African-Americans at all. A store which might not allow black customers, but have black employees, might not even have the facilities at all, to the employee's detriment. In an interview, Dale Morgan recalled his summer job at a luggage store, being told:

Boy... if you have to go to the bathroom, you can't go to that bathroom back there, what we use. You have to get out and go to the cafeteria down the street.

Similar impediments were regularly experienced by domestic staff who worked in white homes where their employers wouldn't allow them to use their bathrooms either. Of course employees were then often chided for needing extra time to use the restrooms....

This lack of facilities also impacted day-to-day lives, as the limited number of facilities was something that had to be planned around, especially for families with small children. As recalled in an interview of its impact on childhood and teen outings, a respondent recalled:

When you are in grammar school, you don’t be going to trips and stuff like that and to games. And in high school we go, you know, we would take part in things like that. And when we went… you were not allowed to use the restroom. When they’d stop at a filling station you could get a soda because they were out front in a box. But you couldn’t use the bathroom. So you would wherever you were going, you had to be prepared to hold it from one place to another. Well at the time, when you’re young and you need to go to a bathroom… that’s the time when it really hurts.

Of course, even when there were facilities, at times they were all but unusable. Although a novelistic depiction, Ernest Gaines' The Autobiography of Miss Jane Pittman paints a stark picture of how little care might be given to the maintenance of such facilities:

up to a year ago they didn't have a fountain there for colored at all. They didn't have a bathroom inside, either. White, yes; but nothing for colored. Colored had to go outside, rain or shine and go down in the basement. Half the time the bathroom was so filthy you couldn't get inside the door. The water on the floor come almost to the top of your shoes. You could smell the toilets soon as you started downstairs. Very seldom a lady would go down there because it was so filthy.

So anyways, to wrap things up, there is simply no definitive answer to this in an overarching way. The underlying intention of ensuring separation between the races, or more clearly, ensuring that black people were kept out of white spaces, manifested itself in a variety of ways, depending on circumstances. But it can be said plainly enough that there was far greater concern with the issue of racial separation, and whether gender separate bathrooms were instituted for black people was a secondary concern.

Sources:

Abel, Elizabeth. "Bathroom Doors and Drinking Fountains: Jim Crow's Racial Symbolic." Critical Inquiry 25, no. 3 (1999): 435-81.

Feagin, Joe R.., Picca, Leslie H.., Thompson-Miller, Ruth. Jim Crow's Legacy: The Lasting Impact of Segregation. Rowman & Littlefield Publishers, 2014.

Guffey, Elizabeth. "Knowing their space: Signs of Jim Crow in the segregated south." Design Issues 28, no. 2 (2012): 41-60.