There aren't anything like reliable Soviet public opinion polls, so we don't really know what they felt about such things, or many things. "Public opinion" means something different under a totalitarian regime in any case.

But in general, the Soviets saw their development of nuclear weaponry as a defensive mechanism. They were surrounded on all sides by US allies, many of which hosted US nuclear weapons or US support bases. (Here is a map I made of US nuclear bomber deployments through 1958. This does not include sea deployments, tactical nuclear weapons, or foreign missile deployments.) The US was the only country to ever use nuclear weapons in war, and until the 1970s its arsenal far exceeded that of the Soviets'. The Soviet people felt under the gun from the origin of the country onward. They could, with some justification, claim to not be the instigator of really any of the Cold War technological excesses — they were always trying to catch up. (Their one area of firsts, the early Space Race, was the result of them concentrating in long-range missiles a little bit earlier than the Americans. So in that area they managed to close the gap, towards the beginning. The Americans were of course also researching long-range missiles, but they also were putting much more research into long-range bombers, which were effective for them as a nuclear delivery system because of the aforementioned forward bases.)

During the 1980s and the period of perestroika, one does find some significant opposition to the arms race, to the military build-up, and so on, as one did in the rest of Europe and in much of the US. One sees very similar attitudes expressed globally about these things, as the threat of total nuclear war seemed particularly acute.

These are generalizations, though, and no doubt don't capture a lot of nuance. Spencer Weart's Nuclear Fear: A History of Images does touch a bit on Soviet attitudes, though again it is limited by the source base.