In the absence of active resistance from an American-led coalition, the Soviet Empire would have been free to expand. The first victims of this expansion would have been European. However, it would not be long before Communist governments were installed in much of what, in our time line, was known as the Third World. Indeed, as the impetus for the expansion of empires often comes from the periphery, most of the invasions, occupations, coups d'etat, and their like would probably have been the work of the leaders of satellite states. (Consider, if you will, the role played, in our own time line, by Cuba in Africa and Latin America in the 1970s and 1980s.)
Life in the peoples' republics of Asia and Africa would, in all probability, have much in common with that suffered by the people of China in the 1950s and 1960s. That is, there would be enormous deprivation, punctuated by periods of dire famine and crises comparable to the Cultural Revolution and the Great Leap Forward. Enough people would survive, however, to change the way that American's viewed Asia and Africa. Whereas the Americans of our own time line viewed the Third World as an arena in which they competed with other people of mostly European descent (i.e. the Russians) for the hearts and minds of people of color, those of this alternate reality would increasingly equate Asians and Africans with the political system of their conquerors.
The presence of large numbers of European refugees would do for America as a whole what, in our time line, Cuban refugees did for Miami. That is, there would be a sufficient number of people telling tales of oppression to temper the natural naïveté of Americans and counter the efforts of those who sympathized with Communism. Indeed, one can imagine a cohort of European intellectuals comparable to the one that, only a few years earlier, had fled National Socialism and the consequent displacement of members of the first wave by those of the second. This, in turn, would have done much to change the character of American intellectual life and, in particular, the composition of university faculties.
At the same time, the reduction of "the West" to the United States (and, one presumes, a closely connected Canada) would increase the degree to which Americans were conscious of the uniqueness of their nation. This, in turn, would drive a stake through the heart of the "liberal universalism" that played such a powerful role in American public life in the post-war period of our time line. In other words, rather than thinking that "most people are basically like us and are on their way to enjoying the same sort of political and economic system that we do," Americans of this alternate time-line would be more likely to think, "we are fundamentally different from everyone else in the world."
If these things happened, it would take much of the wind out of the sails of the civil rights movement of the 1950s and 1960s. It would also prevent the proposal, let alone the passage, of the Immigration and Naturalization Act of 1965.