People have given you good advice about how to get to the Atlantic Terminal LIRR station in Brooklyn.
From there, you could get to your destination one of two ways. You could either take the subway from the Atlantic Avenue/Barclay Center subway station or take a city bus.
The subway is probably faster, but it does require you to change trains within the subway system. Take the R train in the direction of Manhattan 2 stops to the Jay Street/Metrotech stop. Walk in the direction the train was moving ALL THE WAY to the end of the platform and take the VERY LAST escalator. (Be very careful. It s easy to take the wrong escalator at this station and end up on the street outside of the subway system, and have to repay to enter. The signs are a little clearer these days, but I still see people doing this incorrectly and ending up on the street.) Once you are at the top of the escalator, follow the signs for the F train in the direction of Manhattan. Take the F train one stop to York Street. It s a short walk to 55 Washington Street. (Another thing to be aware of - the Atlantic Avenue/Barclay Center Station is HUGE and complicated. The section of that station where the R train stops is all the way at the other end of the Station from where the LIRR trains come in. It s not too hard to do this, but you will do some walking underground. Just follow the signs or ask someone for directions.)
If you want to take a city bus instead, it s a bit simpler. Take the B67 bus from Flatbush Avenue, right in front of the Atlantic Terminal. If you get the buson the same side of the street as the train station you ll be headed in the correct direction. (Just know that you ll need to buy a metrocard before boarding the bus. You can buy a metrocard at the entrance of the subway. Or you could have $2.75 in exact change coins only, no bills.) The bus will take you right in front of the York Street subway station, where you'd get off the F train if you had taken the subway.
Even though the bus ride is simpler, and the subway requires you to change trains, the subway will usually be MUCH faster.