The verb doesn't need to agree with the noun "the brave", it needs to agree with the noun phrase "none but the brave". Consider:
None including the brave ____ the fair
The construction is the same and the requirement for singular/plural is the same, but now it is more clear that the subject of the verb is not "the brave".
I think this reduces the question to the (well-known) issue whether "none" takes a singular or plural verb. The answer is that in common usage it can take either, some people are more pedantic than others about cases where it "must" take one in particular, and monographs on the subject are readily available via your favourite search engine.
Furthermore, "the brave" could be used to mean the plurality of all brave people, or (more rarely) a single brave person. Therefore "The brave deserve the fair" and "The brave deserves the fair" could both be correct.
Mind you, the latter to me sounds like it's talking about a Native American warrior rather than an imagined brave person. But as in the source of the quotation, if you introduce a particular brave person into consideration then you can (at risk of sounding very lofty) refer to that person as "the brave".