This was a very real concern with a very solid basis. Because of the widespread nature of the Soviet nuclear arsenal at the time of the collapse, there were some 35k nuclear weapons scattered across Eurasia, including missiles in Ukraine, Kazakhstan, and Belarus. Given the chaos of the period, it makes sense that the international community was very concerned about the possibility of even one of the warheads from one of the thousands of launch sites/delivery mechanisms falling into the hands of terrorists or criminals. Many of those sites were poorly defended, especially as the government was imploding. The loyalty of guards could not be guaranteed, nor could their willingness to stay in isolated outposts guarding nuclear materials when there was such upheaval in their country. There was no longer a strong centralized authority to provide a significant deterrent against nuclear thievery. According to one Russian military investigator at the time of the collapse, taking highly enriched uranium from a Russian nuclear facility was, "easier than taking potatoes." Added to this was the possibility that internal actors, such as rogue members of the government or military, might take it upon themselves to steal nuclear weaponry and turn it towards their own ends. In short, the threat was very real. As such, helping to curtail the possibility of nuclear theft was a key part of the Nunn-Lugar program aimed at assisting the former states of the Soviet Union in denuclearizing and/or securing their nuclear weaponry. Fortunately, no such theft ever occurred (as far was is known).
Source: What Happened To the Soviet Superpower's Nuclear Arsenal? - Graham Allison