The quote is honestly awkward to me as well, but WH Auden was a poet and you have to let go of traditional grammatical structure to read poetry in English. As for your correction, you are perhaps missing "to": " Many a species succumbs to fate..." To read it as two clauses of one idea: "Fate succumbs many a species": many species go extinct. "One alone jeopardizes itself": if someone is alone they're in danger. The idea seems similar to a quote from Tecumseh (a famous American Indian in the United States): "A single twig breaks, but the bundle of twigs is strong." But that's my take. Please don't try to learn English from quotes by poets.

I'm not sure if you were asking about the meaning of "one alone jeopardizes itself", and as far as I can see, the other answer didn't answer that correctly. I think it means "Only one species puts itself in danger". It is referring to man, and I think it is referring to the way that we are destroying our planet and risking extinction.