The same way you try to combat any other inaccuracy on the Internet: more eyeballs. If you see something that isn't right, write a comment or point it out in chat. If something seriously bad is happening, flag it for moderator attention.
Winter Bash is supposed to be fun, and I really don't want to kill it. But Winter Bash 2015 lasts for 3 weeks, and poor quality content that may slip through during this time might remain with us for years. A bit like a hangover after a fun party, except that it lasts much longer. I wouldn't want our site to be judged by sub-par content that came from greed for moar hatz.
But low quality posts, and off-topic mischief are not.
Here's a list of the common hats that may cause mischief and how to spot it:
Timey Wimey: this hat is received by editing old posts, so you may find users making pointless edits just to get it, if you see some that look suspect, feel free to roll it back / flag.
Thalia & Melpomeme: These hats are received from reopening/undeleting and closing/deleting respectively. You may find users making bad calls simply for the hat. Doing so can result in bans, so actually do it legitimately.
Greeter: Like Timey Wimey, you may find users editing new posts with pointless changes.
Hairboat's Revenge: This is the one that you'll find clogging up chat transcripts a lot. This badge is earned by commenting on Jon Ericson's only CR post, these comments are usually cleared by CMs or Mods soon, but nonetheless.
Speedy Delivery: This badge is earned by answering a question less than half an hour after it was posted, and subsequently earning at least +3 and being accepted. As such, you may find users posting a small amount of review within that half an hour, and doing the rest afterward.
Living in the Future: This badge is earned by selfie-answering a pre-WB post of yours, so you may find users answering their own old, potentially dead posts with new stuff. This is not necessarily mischief, but can become so if posted with little more than required as on-topic.
Auld Lang Syne: Posting a starred message with twelve hours of UTC NYD that gets starred. While not an issue now, chatrooms may find themselves star-flooded on the night.
Sun Wukong: Here's the one to watch out for: A hidden hat earned by a meta post with -5/+5. Users may post crap on meta that gets downvoted solely for the hat. Some SE sites have allowed this, others have not. This may be something that gets Meta'd later.
Vote Early/Vote Often: Earned by voting a considerable 250 times in 7 days.
As the vote allowance for 7 days is 280, this is nearly equivalent to capping every day. As such you may people wasting upvotes and downvotes where they normally wouldn't.
I'm Batman: This hat is earned by editing a question that gets reopened after doing so. A major off-topic cause on Code Review is questions not including code. Some questions do so after being closed, and they are subsequently reopened. But some users get in while the question circulates in the Review Queue and make pointless edits.