From the New York Times:
"The rise of that [jazz] scene — which, like its borough, is an assemblage of enclaves — has been one of the most significant developments for jazz in New York in recent years. (Every bit as significant as the Brooklyn rock explosion of a few years ago, with which it shouldn't be confused.) Through a growing network of low-rent spaces mostly booked by enterprising musicians, Brooklyn has assumed a vital role in the city's larger jazz culture. And the music has been a boon for listeners of all kinds, including those who have to cross the East River to hear it."
The article does a good job of covering jazz spots across the borough.