Assuming you're asking about those two stations (and not York Street or Clark Street) because you're coming from the A or C lines, High Street is closer to the bridge entrance.
I would take the 2 or 3 trains to Clark St and walk to the right when you get to the street from the elevator (west along Clark St.) just to see all the beautiful brownstones and tall trees and completely quiet.
At the west end of Clark St (and all the east-west streets) is the Promenade with stunning views of New York Harbor and the Wall Street District and the Brooklyn Bridge itself (not to mention the Empire State Bldg, and the Statue of Liberty). You can then walk through the area to Cadman Plaza and the steps or ramp leading up to the center walkway of the Bklyn Br. When up there stay on the walking side, not the bicycle side; they are speed demons up there :^)
Also the Court St Station of the R train is a good starting point if you want lunch first. Look at this map for more clarity:
Finally, when you get to Manhattan you are on the east side of City Hall, near the Brooklyn Bridge station of the 4,5,6; also the City Hall Station of the R; also the Park Place Station of the 2,3:
Thanks for all of the info.
For us, the F line to York Street would definitely be the best starting point. Then call into the Brooklyn Roasting Company cafe on Jay Street for excellent coffee and snacks before walking down to Empire Fulton Ferry Park for the iconic views and Janes Carousel before crossing the Bridge.
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