Cross the bridge by walking, preferably at dusk and you'll see the most incredible views of New York. The best things in life are free (And I won't give'em to the birds and bees ;))
This bridge it is worth visiting at least two times. One in the daytime and the other by night. Once you have crossed it, I recommend you to rest and have a look from the little stone beach that you can see in the pictures.
Another way to do this is to take the subway to the brooklin stop and walk through it towards manhattan. That way you can enjoy city's skyline and you can take a break in the park next to the City Hall. And the Courtroom (seen in a LOT of movies) is close by to!
I was walking around Brooklyn Hights, from the promenade you have a great view of Manhattan and after I crossed the bridge. Here some pictures.
MUST DO !
Great Experience! Beautiful views of the City :)