Livery badges were used to identify the followers of a great man, such as a Duke (which Prospero is) or a King (which Alonso is), so when Prospero asks if "the badges of these men . . . be true," he's asking who Stephano, Trinculo, and Caliban belong to. Prospero may be using the word "badges" as a metaphor for the drunken state of Stephano and Trinculo, but they may indeed be wearing the badge of Alonso, King of Naples. Caliban is almost certainly not wearing a badge of any sort.
Embroidered Livery Badge:
White Boar of GloucesterCopper Alloy Livery Badge:
White Boar of Gloucester