Of course, and this is not a pattern confined to the United States. The Cold War, and the broader ideology of 'anti-Communism' grealty benefited right-wing political parties in the UK, Australia, New Zealand and Germany (some of the more prominent examples), as the left (especially in the 1940's, 50's and 60's) could be portrayed rather easily as soft on communism.
Could they though? Let's look at Britain. Until 1955 the Labour Party was led by Clement Attlee who had led a government that had sent troops to fight Communists in Greece and Korea, had played a leading role in the establishment of NATO, had developed nuclear weaponry, had prioritised military spending over social spending, and had systematically expelled parliamentarians suspected of being fellow travellers. After 1955 it was led by Hugh Gaitskell who had built up profile within the Party largely because of his strong anticommunist views, who had been the Chancellor who had prioritised military spending over social spending, and who's closest advisor was a man who used the Durham Miners Gala as an annual anti-Soviet propaganda show. This was not a political party that could be credibly accused of being 'soft on communism'; indeed the only people who thought it was were gin-addled Daily Telegraph-reading colonels and the like, and such people were (it is fair to say) rarely known to be swing voters. Labour lost elections in the 50s because the Tories had been lucky enough to squeak into power just as the postwar economic boom got going, not because of the Cold War.
Two excellent examples, though I wouldn't say the party can be defined by its leaders so easily; least not by such a consensual leader like Attlee who is almost the opposite of a dominating "one-man-party" type - did the Tories not try to associate someone like Bevan with being soft on communism, as well as hyping him as the man really in charge of Attlee's Labour Party?
How far though in your opinion is what you say true of UK Labour in the 60s and 70s - the height of the Cold War - when the Militants and so on were starting to rise?
In Australia the Petrov Affair is in 1954, The Split is in 1954-5 and because of AV immediately benefits the right through DLP transfers if not through directly associating the ALP with Communism. Of course Menzies had already started playing the communist card with the 1950 ban and 1951 referendum (arguably you could say he started it ten years before that, but that was in response to the Nazi-Soviet pact)
