Not much, as it turns out. According to a New York magazine feature by Rachel Friedman, The New Yorker pays just $460 for a 36-line poem. Does anyone pay more? Not according a survey of handful of prestigious journals. The current going rates, according to New York:

$460 for a 36-line poem: The New Yorker
$75 a poem: The Paris Review
$25 a page: Plough-shares
$10 a line: Poetry Magazine

But, you say, poets make all their money from teaching! Some do… but New York also points to a tiny job market for teaching poets—just 750 positions in all.

A few more interesting numbers (royalties! cash for the Nobel Prize) are collected here.