After the Cold War, there was no longer a Soviet Union. Therefore, I will assume that this question is meant to refer to the time during the Cold War.
We cannot know for certain how the Soviets as a people felt because there were no opinion polls taken in the Soviet Union during this time. However, we can generally say that the Soviet leadership was afraid of the Americans. We can also say that they tried to instill a fear of the Americans in their people so that their people would be more likely to toe the party line.
The Soviet leaders were as aware of American capabilities as possible. Early in the Cold War, they knew that the United States was ahead of them in many types of technology. Later on, after the Soviets had a huge nuclear arsenal, they knew that the Americans could destroy them as easily as they could destroy the US. Therefore, they had a healthy fear of the US.
The Soviet leaders tried to make their people fear the Americans as well. They wanted their people to believe that the Americans were aggressive and volatile. They hoped that this would keep their people in line. Therefore, we can at least assume that the Soviet people feared the Americans as much as the Americans feared the Soviets.