What is the effect of the author never explicitly stating the family's race?

Gordimer structures her story as a fairy tale, as the title "Once Upon a Time" indicates. Fairy tales are usually universal rather than specific, mythic rather than historically grounded. They are set in an unspecified past ("once upon a time") and an unspecified place: a generic forest or a generic castle, for instance, rather than a specific forest or castle that we could locate on a map. This puts fairy tales outside the realm of real history.
Gordimer does not specify race, so she can make clear that the problem she is discussing transcends South Africa's situation with apartheid at the time. These are problems that can emerge anywhere, at any time, if one group of people fears, excludes, and tries to wall out another group. The larger message is not that South Africa alone is "bad," but that it is destructive to everyone, no matter what the place or time, to live in a society dominated by fear of the Other.

Although Nadine Gordimer was a famous writer-activist from South Africa, she chose to make this particular story "color-blind." Essentially, she chose to omit the race of the family she wrote about. 
While this may seem curious to us, Nadine's omission may have been intentional; in fact, the omission has a stunning effect on readers. It shows that the race of the family is not as important as the fear they live under. We are led to question why the family lives in such fear and why frequent riots occur in the city.
We are also led to speculate as to how the social fabric can unravel with such frightening ease.
The race of the housemaids, gardeners, and rioters are also not explicitly stated. Gordimer simply refers to these people as "people of another color." By not stating the race of the groups in conflict, Gordimer may be making the point that race-driven conflict is dehumanizing in nature.
Like the city rioters, the housemaids and gardeners are purportedly members of "another color." These domestic maids and gardeners do not share the same racial identity as the families they serve in the suburb. Yet, they share the same distrust of the city rioters that their employer families do. 
When the mother of the family orders her housemaid to bring out bread and tea for the loitering masses outside their gated house, the latter refuses. She proclaims that those outside the gate are "loafers and tsotsis" and are not to be trusted. After all, they will only "tie her up and shut her up in a cupboard" for her kind deed.
Despite sharing the same racial identity as the loiterers outside the gate, the housemaid refuses to trust her safety to their hands. Accordingly, what Gordimer may be trying to say by omitting an explicit reference to the race of the family is potentially this: it is not our racial identities that are important but, rather, the question of why we are in conflict with another.
Gordimer chose to focus on the issues of security and equality; to her, these issues transcend race and nationality.