Bandit morale is basically how content the bandits are in their camp. If their morale is low they will stop playing nice and go back to their old ways. If it's high they respect the MC's rules and play nice with the people of Bareshade.
You increase it with a couple of quests from Jester. One of them is repeatable, "Bitch for Bandits." You either allow them to sate themselves with your body, or, if your Hero score is high enough in Bareshade you can recruit a fanboy to take care of that for you (but you have to go to the camp to make him do it).
You can check the bandit morale by clicking on the camp while on the map.