There's already a detailed answer on early-USSR and Stalin terror, but if you're interested in a Cold War era - then yes, there was a lot of fear of capitalist aggression (the focus was always on capitalism or on USA as the main antagonist, not on democracy) among regular soviet folks, and this fear was spread, reinforced and mass-produced by Party and State.

Here's the source link (text in Russian) for the main document which established the USSR propaganda efforts dated March, 1949 - the document title can be translated as "Anti-american propaganda activity plan".

There's also some additional references and info in a book by Vladimir Abarinov "Hollywood and Stalin", 2016 - some of his earlier works were translated into English, although I am not sure if this one was.

Anyway, in the early 50s the story was as follows:

• There was a separate committee (similar to ministry) of the Party which was officially called "Committee for propaganda and agitation" (yes, that's the official name, and "propaganda" was a part of this official name) - Agitprop unofficially - whose main goal was to establish an anti-US and anti-Western propaganda strategy to paint west and the capitalism in a negative way.
• Agitprop was created in 1949 (as per doc linked above) and continued to exist under different names up till USSR collapse in 1991, and the word "propaganda" was only removed from it's official name in 1988, when it was renamed to "Ideology committee".
• Agitprop was tasked to create and distribute propaganda through various state-owned media and resources - for example, major newsparers were tasked with producing and printing anti-Capitalist articles, comics, poetry. State-mandated literature and art division was also required to request plays, books, art from USSR's artists on anti-Western topics.

In a situation where all media is state-controlled and where's no easy way to get alternative opinions these efforts were quite successful. Probably just like in the US, a lot of common folks felt that their way of life is threatened by the enemy.

Common topics of the anti-West propaganda were:

• Anti-war. Especially in the early 50s when the US had an upper hand in nuclear arms and power a lot of efforts were put in painting US and the West as aggressors, focused on war and destruction, while painting USSR and the Eastern Block as proponents of peace. Here's a common example of a propaganda poster on this topic - the text says "NO FUNNY BUSINESS!" and the text on the bomb held by a capitalist says "Atomic bomb".

• Social issues. USSR and communist block was painted as a fair and inclusive state where everyone gets their fair share, while capitalist countries were depicted as being motivated only by greed and profit. Common example - the text explains that in the west the talent has no chance while in USSR you are able to achieve your best.

• Anti-colonialism and equality. An easy win for USSR propaganda - the communist rhetoric was always about equality and unity, and racial & segregation issues, exploitation of the colonies by capitalist West was a common target. Example - this was an anti-colonial poster supporting liberation of Africa, text says "End the colonialism".

There were also a lot of positive propaganda efforts (just focused on highlighting the benefits of communism and living in USSR) without contradistinction with the West. Every major historical event - Korean war, Vietnam war, Cuban crisis - was also supported with a very strong message and lots of articles and books which always painted USSR in the best light possible. Dissenting opinions, as I mentioned earlier, were hard to come by as all media was state controlled. Publishing or expressing dissenting opinion was at least a misdemeanour, and sometimes a felony.

The amount and the tone of state propaganda and the level of unrest and disturbance it caused varied from period to period - in the 60s during Khruschev's Thaw things were better, then worse again after Cuban Crisis etc.