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.

Quite often, the right choice is not to dignify it with a response. Consider whether or not the other people who were copied in will care. If not, responding isn't necessary, and makes you look like you care far too much about something trivial. Similarly, if they already have a more negative opinion of the person who sent you the email than they do of you, then responding isn't necessary. It's only worth responding if the other people respect the person who sent you the email and care about the issue at hand. These two conditions are often not satisfied.

If it is important to respond (e.g. because you'll get incoming if you don't), it's generally best to respond not to the email in question, but politely to the email containing the evidence, copying in the relevant people. A friendly "thanks for doing this after all in the end - much appreciated" will generally suffice. Everyone else will get the point, and the person involved will either have to let it go themselves or look foolish.

Regardless of the setting (i.e. academia or corporate workplace) there is only one way to handle this: Professionally.

That leaves you with two options:

Option 1: Ignore it and move on. Maybe they were having a bad day. Maybe they aren't great at communicating. Maybe it's a once off. Either way, there's no personal damage to you and some things just aren't worth escalating. If you see the person, talk to them. Say something like "Sorry for the misunderstanding, but I thought from your other email that it was no longer required. Thanks for completing it for me". This gives them the chance to also respond and offer an apology of their own, if they are so inclined.

Option 2: Respond to his email directly, with something similar to the format suggest be Geoffrey Brent in his answer. Do not, under any circumstance, reply all and list your grievance, why you're right and demand an apology. Respond directly to the email, CC or BCC in your supervisor so that they are aware, and respond in a professional, succinct manner. And then leave it at that.

Responding in a Reply All email, listing why you're right and demanding an apology is not going to do you any favours. Keep it private and keep it professional. Apologise for any misunderstanding based on the previous email and leave it at that.

If they decide to continue, that's when you escalate with your supervisor and possibly further.

Most likely though, no one other than you is concerned or even paid attention, and chain emails to everyone merely make the sender look bad. The last thing you want to do in these situations is air all the dirty laundry in public. And if things do go pear-shaped, refusing to be drawn into drama and responding in a polite, measured, professional manner puts you in a good light.

Either way, chalk it up as experience and move on.