Let's bring in a little context.
"He is to find consolation in exercising the highest functions of human nature. He is one, who raises himself from private considerations, and breathes and lives on public and illustrious thoughts. He is the world’s eye. He is the world’s heart. He is to resist the vulgar prosperity that retrogrades ever to barbarism, by preserving and communicating heroic sentiments, noble biographies, melodious verse, and the conclusions of history."
--Ralph Waldo Emerson
The speaker suggests that his subject should avoid succumbing to the temptations presented by the cheap thrills provided to him, which lead him back to a more barbaric state of being, instead aspiring to something better.
To resist here is to avoid temptations/inclinations.
Vulgar prosperity here is presumably vulgar in its lack of artfulness and illustriousness, by being common and unenlightened by history and the arts.
Ever suggests that succumbing to the aforementioned vulgar prosperity will always lead back to barbarism.
Retrograde here means to reverse or revert, working with the notion that barbarism is a primal human state.