Almost anything, and more or less. Some Judges -- mainly Mirror at this point -- take penalties from teams that lose their games, and occasionally Judges will temporarily claim Blanks for tasks they need done, but otherwise, they mostly stick to the Arena. There are a few exceptions, and I'll get to those shortly.
As to what they need enforcers for...mostly it's to keep watch on us when they can't, give out tasks in exchange for luxuries and items, and stir conflict when possible. There's a list of rules they've emplaced, most of which are relatively pointless. We're not allowed to share food or supplies between non-partner teams without a fair trade, sleep over at a dorm that isn't your partner's, stay out after curfew, punch walls -- it sounds strange, but people used to do it to test innate abilities like strength and stamina -- kill outside the games, or visit a place we no longer have access to, which I'll also get to shortly.