Probably after LBJ at least. Kennedy proved himself to be tough on Communism and LBJ followed the containment policy through Vietnam. Although the one Democratic president we had in the final decades of the Cold War (Carter) was voted out of office because he was was not seen as tough enough on the Iranian Shah, who would probably be considered closer to a Fascist than a Communist.
or too soft on the Islamists that replaced him? Wasn't it Eisenhower that had spies overthrow the popular left-wing government there in favor of the Shah over fear that Iran wanted to join the USSR or at least the Warsaw Pact? Though, most likely, Iran was probably going a very similar route as India and would probably by non-aligned, being opposed to both Russian and Western influence (Iran was occupied by both the UK and Russia after WWI).
The point I am trying to make is that every president was about as aggressive against Communism (save for maybe Nixon, Ford and Carter but that was probably because of the problem with Vietnam and the resulting demoralization of our relations with Communist countries) the difference was their strategy. Reagan probably had the best strategy of letting them come into our back yard but then outspending them so that we had both the economic and geographical advantage. However, as you way know, his tactics in fighting those proxy wars were unethical though they couldn't be proven to be illegal.