Place chocolate in a heatproof bowl set over (not in) a pan of gently simmering water. Stir just until melted.
Line a baking sheet with waxed paper. Insert a popsicle stick in one end of each banana piece. Dip banana, one piece at a time, in chocolate, spooning on additional chocolate to cover.
Sprinkle each banana with peanuts, and set on prepared baking sheet. Refrigerate until chocolate is firm, 20 minutes, or up to 3 days.