A verb takes its infinitive form when combined with auxiliary do:
Do ghosts exist?
Does God exist?
Notice that exist does not agree with the subject. The choice of do or does agrees with the subject. Do and ghosts are both plural; does and God are both singular.
Since this is a question, you need the inverted form, where auxiliary do precedes the subject and the main verb follows the subject:
Do any other approaches exist?
In English, we don't usually use the verb exist. Usually we use the dummy subject there with be, like this:
Are there any other approaches?
Yes, there are other approaches.
That's probably how you should say it. Usually we say exist to talk about whether something is real or a superstition, for emphasis, or to speak precisely and technically, as in mathematics. By using exist when searching for a possible method ("approach") for doing something, you make the listener wonder if you mean something like what people ordinarily use exist to mean.
However, for that reason, saying exist where be is customary makes your point more forcefully. In an appropriate context, if you say "Do any other approaches exist?", people will understand this as a rhetorical question: that is, you mean "Stop searching for something that might not even exist! Let's choose the approach that we already know."
Do there exist
It is within the limits of grammaticality to combine dummy there with exist, but it sounds awkward:
Do there exist any other approaches?
Most people perceive this as clumsy English. Often in English, we do get clarity by combining multiple ways of suggesting the same thing, but combining dummy there with exist sounds overly wordy. I don't know if there is a principled explanation for why. English works more by customary phrases than by rules, so there might not be a rule here. In mathematics, does there exist does not sound as clumsy, because there exists is a very common phrase in mathematics.
Does there exist a greatest prime number? [Ordinary and clear in mathematics.]
Does a greatest prime number exist? [A little strange-sounding in mathematics.]
Does there exist a credible presidential candidate? [A little strange-sounding in everyday English.]
Does a credible presidential candidate exist? [Emphatic everyday English, suggesting doubt about something ordinarily taken for granted.]