As an AP World History student, we sort of flew through this topic, and I found the entire time period of the Pax Mongolica very interesting. And as I know that underpopulation of an extremely large land empire was one of the major reasons for Mongol decline, it doesn't seem right that the Mongols would be able to efficiently make the silk road safe for trade.
One of the most basic ways in which they reduced banditry is that the former bandits were now the guards. One of the biggest risks if you were travelling across the Central Asian steppe was attack by the various nomadic groups that populated the area. Since these groups often got subsumed into the Mongol armies the risk of attack declined sharply. Furthermore the re were regualr patrols of the trade routes by the Mongol army. The extreme mobility, efficacy and ruthlessness of the Mongol armies meant that they were very well suited to dealing with any bandit threats that might arise along the long roads.
Furthermore the Yam system meant that the roads were rather well manned. The Yam was a series of way stations complete with supplies, fresh horses, lodging, etc Thus the Mongol government was very present on the roads that criss-crossed its empire. This system's primary purpose was to allow the rapid transit of messages across the empire, and its efficiency appears to have been remarkable some reports indicating it allowed messengers to travel 200-300 miles a day. This admirable efficiency would have meant that any local disturbances could be quickly relayed to the relevant authorities.