What is/are important are not your mistakes, but how you respond to them.
Which option makes more sense grammatically?
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Neither of them make sense, grammatically, because mistakes is plural and how you respond is singular.
If we are forced to make a choice between the two, the best option would be to use is, because it's less of a stretch to consider mistakes as a collection (that is a singular set of mistakes), than it is to try to justify writing What are important is (or are) how you respond . . . .
I would personally not be comfortable writing mistakes is in this context, however, at least not in and I'm a pretty liberal grammarian (sometimes even a grammamatician). In conversational speech, it's the kind of thing many of us do without many others noticing or caring much.
The better option would be to re-write around the problem, for example:
It is not your mistakes that are important, it is how you respond to them.