From JFK you'll probably want to take the B15 bus, the A train, or the LIRR. All of them will get you very close to Crown Heights.

Just be aware that some of the neighborhoods around Crown Heights have among the highest crime rates in the city. I don't really know the area that well, and my guess is that during the day most of it is perfectly fine as long as you maintain normal street smarts. But i don't have a good sense of whether, for example, transferring from the B15 to the 3 train in East New York is a reasonable thing to do.