Yes, there were immense fears about the security of the Soviet arsenal, and fears that their nuclear scientists might sell their services to the highest bidder. The fall of the Soviet Union was in many ways the biggest nuclear security crisis of the century, because not only were the guards under- (or un-) paid, but the facilities in many cases lacked sufficient inventory control to even know if something had gone missing.

One of the immediate approaches, which is sometimes cited as one of the most significant "payoffs" in terms of arms control and nuclear security, was the Soviet Nuclear Threat Reduction Act of 1991 (usually called the Nunn-Lugar act after its co-sponsors), which basically allowed US money to be spent on upgrading Russian nuclear security and dismantling nuclear facilities in the non-Russian former-SSRs. There were other legislative pushes meant to make it easy for Soviet scientists and engineers to emigrate to the United States (as opposed to, say, North Korea), which were part of this attempt to prevent a massive leakage of WMD information broadly. And there was the Megatons to Megawatts program in 1992, through which excess Russian plutonium was purchased and down-blended into civilian nuclear reactor fuel and used to make electricity in US nuclear reactors.

All of these were large, expensive, bipartisan efforts that only had a chance of succeeding because people in the US in particular were terrified of the possibility of the massive Soviet nuclear arsenal becoming more dispersed.