Keep in mind that there were two "Red Scares" in our history, one essentially soon after the communist revolution and the formation of the Soviet Union in which we aided "White Russians" in the ensuing Civil War, and conducted the Palmer Raids in 1919, deporting hundreds of Russians we deemed as people who could foment a revolution in the US, and a Second Red Scare in the late 1940s and early 1950s. So our assumption in the First Red Scare was that the immediate Soviet goal was to spread revolution to the US, who at that time was the largest industrial economy on Earth.
The Second Red Scare, as detailed above, rested on two assumptions: 1) that the Soviets were aggressively expanding around the globe and would only react to the threat of force, and 2) that they had successfully infiltrated the US government and military, an assumption which led to McCarthy's hearings in the US Senate.
During the Red Scare, Communism was considered inseparable from Anarchy. The Soviet Union was considered the face of Communism and thereby the face of the enemy. Those who supported the party, real or perceived, were believed to advocate the overthrow of the American system of government. This came from the Communist belief that the workers revolution was to be a worldwide revolution--they would take over the world. The Communist anthem was :the Internationale. This belief was augmented by the practice of many in the labor movement. Big Bill Thompson, one of the founders of the International Workmen of the World, fled to the Soviet Union and on his death was buried in the Kremlin Wall as a Hero of the Worker's Revolution. Our belief was that the Soviet Union was the face of that threat, and it needed to be contained. Woodrow Wilson actually sent troops after the Russian Revolution to restore the old order, but this failed miserably. The belief that it was a true revolution led to the containment policy of the Truman and Eisenhower years.
The major assumptions that underlay American foreign policy towards the Soviet Union came from the famous "Long Telegram" written by George Kennan. In this telegram, Kennan argued for a policy of containing the Soviet Union.
The basic assumptions that the US made said that the Soviet Union saw itself as being in a perpetual war with the capitalist West. The Soviet Union, we believed, wanted to expand the communist sphere and destroy the capitalist West in this conflict. We believed that the Soviet Union would use other communist countries as its tools in trying to achieve this goal.
In short, the basic assumptions were that the Soviets were a hostile and aggressive power and that they were bent on destroying the United States and its allies.