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Getting there from Penn Station will be really easy, take the 2 or the 5 train from the Penn Station subway stop. You will get off at Winthrop St. Cost: $2.25. It will take about 40 minutes door to door.
From JFK, you can either take a cab, which will cost about $30, or public transportation. The latter option is really cumbersome and will involve at least 2 long transfers, and will cost about $10.
I'd opt for the cab from JFK, but the subway from Penn Station.
Note that Winthrop Street station on the #2 is a little funny in that the Brooklyn-bound and the Manhattan-bound entrances are not at the same street. When you get off the train coming from Manahttan, you will be at Parkside and Nostrand Avenues, rather than at Winthrop Street. Walk south (with the flow of traffic on Nostrand) one blcok to Clarkson, and turn left; the Downstate campus begins one long block away on the other side of New York Avenue. Coming back, when you get back to Nostrand Avenue you will need to walk TWO blocks (and not one) north on Nostrand Avenue to Winthrop Street where the entrance to the northbound trains can be found.
The subway will be both cheaper and faster, and also less likely to get lost; it will be a rare cab driver in Manhattan who will have any clue whatsoever where Clarkson Avenue in Brooklyn is. If you cannot give turn-by-turn directions (which obviously you can't, or you wouldn't have asked) you should never consider jumping in a cab in Manahttan to go to the middle of Brooklyn.
Also, the 5 train doesn't stop at Penn Station. If you go looking for it, you'll get lost.
^ yep, brain-freeze. Take the 2 train.
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