A staff member was rude to me in an email and made me look bad in front of other members of the staff. How should I handle this?
It sounds as if your co-worker was extremely rude, and owes you an apology, but pointing those things out in so many words is rarely productive. Instead, I would use something like the following template with a reply-all:


  Hi John, there seems to have been some misunderstanding. From your
  email of January 28th (copy attached below), I understood that I was
  no longer required to feed the departmental axolotls - if I misunderstood this, could you please clarify what was intended? 
  Anyway, thanks for fielding it on this occasion. 


At that point, if your co-worker is a sensible person who was just having an off day, they will offer an apology without further prompting. If not, well, your other colleagues can form their own judgement based on the relevant facts.

edit: 

re. whether it's better to respond to the CC list, or just in private: you certainly have a right of reply in the same venue, but whether it's advisable to exercise that right depends very much on circumstances. 

If the alleged omission is sufficiently trivial ("you didn't wash up the teapot yesterday!") then "John" has already made themself look silly just by making a big deal about it, and it's not likely to harm your reputation, so I'd just let it go. The annoyance value to others from prolonging the thread outweighs any need for reputation control; cf. the old adage about wrestling with a pig.

At the other end of things, if leaving the allegation unanswered could harm your professional reputation ("you didn't label the poisons cabinet/feed the research animals/complete the grants proposal due next week"), and if "John"'s allegation might seem plausible to the recipients, then it becomes important to rebut it in the forum where it was made. You don't want a dozen co-workers believing that you're a menace to their health or their work.

Leaving aside your own reputation, it might be important to reply for other people's sake. People take their cues on acceptable behaviour from those around them, and bullies often bully more than one person; for either of those reasons, it can be beneficial to see bullying behaviour promptly challenged. 

Because workplace culture varies so much, it might be wise to seek input from colleagues at your institution - they can probably tell you better than us whether people take your co-worker seriously.