First let me try to answer the question in an "ideal world", where (in particular) set theory is treated like any other branch of mathematics, and then let me discuss how we might fall short of it.

In an ideal world the question of "What is the importance of your field to the rest of mathematics?" would not get asked so often (unless the field is so small that one has never heard of it before) but the question "What is the importance of your work to the rest of mathematics?" would. The reason for this is that the web of connections between different fields is one of the things that makes mathematics most beautiful.

Of course, this does not mean that if someone's work is not important to other fields of mathematics, mathematicians should reject all their grant proposals and deny their job applications. It just means that connections to other fields of mathematics are one possible very strong selling point. There are a number of reasons that a mathematical theorem can be interesting, of which applications are one.

In my preferred subfield (etale cohomology theory, one of the most fashionable topics of the 1960s), and I suspect in many other fields of mathematics, the way I see it is that there is a technical core of the subject, consisting of works that address the most difficult technical problems or advance the methods of the field, and a surrounding periphery consisting of works which attack specific examples or make only small variations on existing techniques. It is these peripheral works that need applications. (Let me clarify that I am not trying to criticize other mathematicians' works - I am thinking primarily of my own work when I think of work on the periphery, though I am proud of some of my applications.) To convince other mathematicians of the importance of works without applications, one must lean on other points (How long a problem has been open, who has tried and failed, the way it connects different subfields of your field).

An additional advantage for those whose work has applications to other areas of mathematics is that mathematicians in those areas will better be able to understand when they talk about the importance of their work. When someone explains to you work in your field, or that has relevance to your field, you are able to more accurately judge its value. For work in another field, even if it's equally good, you may not be able to verify this yourself. For this problem it seems the only recourse is to try harder to explain your work, for instance using analogies with other work.

Now what may be the problem with set theory?

It seems to me that non-set theorists may have a skewed perception of set theory for a number of reasons:

Some mathematicians view math entirely through the lens of finitary mathematical objects.

One reason is that, because set theory is the foundation of mathematics, most mathematicians work with sets at some point in the research, but typically with sets that are almost entirely dead (i.e. devoid of intrinsic set-theoretic interest). Most mathematicians probably work with only finite sets and sets of zero, one, or two different infinite cardinalities. So if you work with sets you work with something that the mathematician has seen, but not that they have seen be interesting.

Some mathematicians are skeptical of both platonist and formalist philosophical views in mathematics, and take comfort in the fact that their research (and most mathematical research) can in principle be reduced to statements about concrete, finitary objects. For set theory this usually can't be done except in a purely formal way.

Perhaps only because I am optimistic, I think the biggest reason is that mathematicians simply do not understand set theory very much. It is not so transparent to outsiders how a zoo of exotic mathematical fauna arises from the simple concept of size. I think that a lot of mathematicans who are opposed to set theory would be receptive if it were explained to them in an appropriate way. One merely has to examine the top reputation of all time scoreboard on mathoverflow to demonstrate that it is possible for a set theorist, by explaining set theory well, to be popular among mathematicians of all stripes.

Probably the best advice for how to do this is to find the mathematicians in your field who have the most professional success, and try to see how they present themselves to a broader mathematical audience. I attended a beautiful colloquium talk by Hugh Woodin which I thought did a great job of explaining the importance of set theory to non-set theorists without, I think, ever mentioning an application to other branches of mathematics.

There are too many B.A.D. mathematicians,Mathematical Intelligencer15 #1 (December 1993), 6-9 freely available version. See especially the paragraph beginning with "I was shielded from mathematical bigotry until" and what follows. $\endgroup$ – Dave L Renfro Apr 28 '19 at 19:42doesinclude the caveats I referred to) $\endgroup$ – Yemon Choi Apr 28 '19 at 22:392more comments