This kind of thing can be very hard to estimate, since the Communist Party obviously presented itself as having the support of the vast majority of people during its time in power. While it's certainly true to say that the average Soviet citizen knew far more about Marx than the average American citizen knows about Adam Smith, Soviet education was less focused on teaching people the ideas of Marx and Lenin and more about applying those ideas to aspects of education.

If you want any kind of broad survey of the popularity of the Communist Party, we have to look to elections. The only free election held in the Russian soviet was in 1990. Opposition parties were not allowed to stand candidates but independents did stand. The Communist Party still won 86% of the vote on a 77% turnout. The indication would appear to be that, during their period of political hegemony, the Party had some level of support from the majority of the population, even if this support came from a lack of alternatives. Possibly more interesting are 1990s Russian election results. In 1995 the Communists took 22% of the vote and in 1999 they took 24%. In 1996, the Communist candidate for the presidency won 40% of the vote. To this day, the Communists remain the largest opposition party in Russia. Even though the Soviet period was widely criticised and discredited in the media and in public discourse, a great many Russians still supported the Communist Party. So, the election results of the 1990s indicate that the Communist Party could rely on the broad support of most people while they were in power, but also that many people's belief in the party was so strong so as not to be shaken by its downfall.