How can I deal with a worker who feels that she is better than others? She has no education, lacks sophistication and the ability to think analytically, yet she consistently mistreats others, sometimes to the point of being out right rude.

I don't supervise her and her supervisor is not very approachable... I find that this person ends up mistreating the individuals I am in charge of supervising.

Any suggestions?

Ask to sit down and talk to her about her behavior. Or ask her to leave the people you supervise alone, or you will have to talk to her supervisor about this behavior.

Is there someone higher up you can go to with this?

I'd talk to her first, let her know that you don't appreciate this behavior, maybe she's not aware of it.

Good luck.

If you are a supervisor, then you supervise, you approach the other supervisor and expect them to act on the problem, Depending on company policy even say something to the other employee if their supervisor does not. ** this will depend on your company culture.
If not after trying to work with other supervisor, go up the ladder.
Part of the duties and problems of a supervisor is doing what is hard at times

You should write down 5 or 6 of these rude and demeaning comments, then ask their supervisor to allow you to speak with them about it in private with the affected coworkers present.

If an employee insists on being rude, being publicly called to explain each phrase and inference in front of the offended people (they should be instructed to remain silent throughout, regardless of what is said), often it will result in self-curbing afterwards.

"Johnny, I have several things here we need you to respond to. First, when you mention you can do X-job better than Cindy here, please explain exactly how you come to that conclusion and support your position."

"When you mentioned to Sandy that Phil was unqualified to do his job, can you explain exactly how you support that? Also, explain to us and Phil how it was important for you to discuss that with Sandy."

"Now, can you provide for all of us here the experience and educational background you have to support your condemnation of other's skills or abilities? Please be specific and I may need contact numbers."

This is done behind closed doors, but also in front of the others affected. It should appear to the rude person that none of you are coming at him "mad" or "out to get him", you present your queries as matter-of-factly as you can muster. If/when he gets defensive and attempts denial, you say something calmly like:

"I'm sorry, you misunderstand. We're not asking if you said or did these things, we already know you did. And that is fine. We are simply insuring there is solid basis for your speaking out this way, and if so, we simply wish to document it. I'm sure you understand. Now, you were saying?"