As you approach the bridge from Brooklyn Heights side, head down Henry Street and you will see an underpass with cars heading through and people walking by. Head that way and swing a right at the first street and you will see a set of stairs leading up to the bridge.
After ascending the two dozen or so stairs to the walking path, the beauty of the view on both New York and Brooklyn. Plus, you get the view of the East River, The Manhattan Bridge, The Statue of Liberty, and the everything else located south and north of Manhattan.
It's kind of odd to be walking on the wooden planked walkway in the center of this bridge while cars zip by a lightning speeds below you. The center posts holding the bridge up are behemoths of stone and wonderment. In the direct center of the bridge, you have the luxury of not having the thick steel cables that support the bridge in your view and the sky is the limit on what is available. Midtown Manhattan is riddled with skyscrapers that from this view point looks more like an extremely good painting set off in the distance.
When reaching the other side of the bridge and entering Manhattan, the effects of Metropolis truly take over. The police plaza (I hope you never visit the inside) is off to your right and a truly remarkably building. But the general attraction of late has been the area of Ground Zero which is only across City Hall Park. But enjoy the walk and try to burn off some the food from either Brooklyn Heights or work up an appetite for the South Street Seaport. And Bring a camera!