At the risk of sounding flippant: akwardly. South Africa by that time had become relatively loathsome for many people across the Western world by that time, yet its enemies were no more palatable to many of those in power, especially with the fact that this was the time of the Cold war tacked on. This can be best seen with Reagan's uncomfortable positioning on the manner, as on the one hand he tried to stop the congress from passing anti-apartheid boycotts of South Africa while trying to appease them with very weak sanctions. The United Kingdom was in hardly a better position, as it not only had close population and familial ties to South Africa, it also had strong economic ties and needed its gold production.

As such, England and the United States engaged in an awkward dance with the Apartheid regime. They went along with the arms boycotts, and made sure that they hit Rhodesia like a block of bricks, while playing more carefully with South Africa. The United States was especially impacted by the Tar Baby doctrine of Kissinger, who believed that the best way to continue was to work within the status quo and use enticements rather than punishments to encourage South Africa to change.

They also took a fairly strong stance against the ANC as well, in large part to its Soviet support and the fact that it was a fairly brutal organization at the time, as best seen through the dichotomy of Winnie Mandela, who spoke favorably as to how the ANC would liberate the country with its necklaces (as in gas-tire necklacing of blacks who were seen as collaborating with the Apartheid regime) and Nelson Mandela, who went through a complete transformation in prison, one which allowed him to become the unifying nation builder he was while in power.

For the Border War itself, you had the fact that Cuba was heavily engaged while UNITA was ostensibly claiming a position much in line with American political values. So basically its South Africa (who you know will somewhat side with you), UNITA (who is quite marketable back home) vs Cuba (public enemy no 1), MPLA (generic store brand communist leaning rebels) all backed by the Soviet Union (public enemy no 2). There was no way the US could remain neutral, and on top of that, South Africa was the only choice.

Sources: Emerson, The Battle for Mozambique, Maier, Angola Promises and Lies, A Profile of Winnie Mandela from the BBC, Berkely, The Graves are Not Yet Full