Short answer: stay on the Manhattan side unless the view is the whole point
The Brooklyn Bridge has two ends and they give you two different trips. The Manhattan entrance is up by City Hall, at the northern edge of the Financial District. Staying down here means you're on the walkway in ten minutes, before the tour groups get organised, and you can be back with a coffee by the time most people are queuing for one. The Brooklyn end, DUMBO and Brooklyn Heights, is where the postcard picture gets taken, and it's lovely, but every trip into Manhattan becomes a river crossing.
If your list has the bridge on it once, stay in Manhattan. If your list is mostly Brooklyn, stay in Brooklyn. That's genuinely the whole decision.
Where the bridge actually starts in Manhattan
The pedestrian entrance is at the top of the Financial District, near City Hall. From John Street you walk north and west for about ten minutes and you're at the ramp. We've written the turn-by-turn version of that walk separately, because the first time you do it the entrance is easy to miss from street level.
The useful part of that geography: the bridge, the Seaport, the 9/11 Memorial and the harbour ferries are all inside the same walkable square. You aren't choosing between them the way you would from Midtown, where each one is a subway ride and a decision.
Walk it early. Really early.
The walkway is shared between people and, depending where you are on it, cyclists moving faster than you'd like. By late morning in the warm months it's shoulder to shoulder and the photos are mostly of other people's shoulders. Go at 7am. The light is better, the harbour is quiet, and you get the cables to yourself for a stretch.
This is the argument for staying near either end rather than commuting to the bridge. Being a ten minute walk away is what makes a 7am walk plausible instead of theoretical. From Midtown it means a subway ride before dawn, and most people simply don't do it.
What else you get on the Manhattan end
Staying at the FiDi end of the bridge means the rest of the day fills itself in. The Seaport is a few blocks east, with the piers and the cobbles. Battery Park and the free Staten Island Ferry are at the bottom of the island. The 9/11 Memorial and the Oculus are a short walk west. Stone Street, which is a block of restaurants with tables in the middle of the street, is about ten minutes south.
And on weekends it's quiet down here, because the weekday crowd is office workers who go home on Friday. That's a real advantage if you're travelling with kids or you want to sleep.
The Brooklyn side, honestly
DUMBO and Brooklyn Heights are worth a morning. The Promenade view of Lower Manhattan is the best one there is, and the waterfront park under the bridge is a good place to let children run. If you stay over there you'll wake up to that view every day, and pay for it in travel time whenever your plans are in Manhattan. We walk guests over the bridge and back in a half day, which for most trips is the right amount of DUMBO.
What a stay at 17 John looks like
We're at 17 John Street, on the Manhattan side, about ten minutes from the bridge entrance. Our suites are laid out like apartments: a full kitchen, a living room, bedrooms with doors. We sell the suite rather than charging per person, so the price doesn't change with the size of your party as long as everyone fits the layout, and for bigger groups we have two and three bedroom suites (a three bedroom handles a party of five). There's no breakfast, but there's coffee in the lounge, and a kitchen if you'd rather eat before you walk.
The gym and the guest lounge are on the lower level, there's a laundry room on site, and the rooftop terrace has lounge chairs, which is the right place to be at the end of a long walking day.
A simple plan for bridge morning
Up at 6:30. Coffee in the suite or in the lounge. Walk north to the entrance and out onto the bridge as the sun comes up. Cross into DUMBO, get breakfast there, then either walk back or take the ferry or the subway. You're home by lunch with the best hour of the day already behind you, and the rest of Lower Manhattan is still on foot from your door.