The one-sentence answer
Pick the Financial District if you'll be moving around Manhattan every day; pick Brooklyn Heights or DUMBO if you'd rather look at Manhattan than be in it.
We're on the Manhattan side, so take that into account. But we send guests over the bridge constantly and we're not going to pretend Brooklyn isn't a good place to sleep. The honest version is that these two areas feel more alike than most people expect. Both are quiet at night, both are near the water, both empty out on weekends compared with Midtown, and both are a walk from the same set of downtown sights. The difference is what you're optimizing for.
What actually separates them
Trains. This is the real gap. Fulton Street in the Financial District is one station serving eight lines: the 2, 3, 4, 5, A, C, J and Z. That means an express to Midtown, an express to the Upper East and Upper West Sides, and a direct ride to Brooklyn or Queens, all from the same turnstiles. DUMBO has the F at York Street, and Brooklyn Heights has the A/C at High Street, the 2/3 at Clark Street, and Borough Hall a bit further in. Those are fine. They're just fewer, and if the F is running badly on a Sunday you'll feel it.
The view. Brooklyn wins, and it isn't close. The Brooklyn Heights Promenade looks straight across at Lower Manhattan, and the Washington Street shot in DUMBO with the Manhattan Bridge framed between the warehouses is the photo everyone comes for. From FiDi you're standing inside the skyline, which is a different pleasure and a worse photograph.
Evenings. FiDi has Stone Street, a two-block cobblestone run of restaurants and bars, plus the Seaport by the water. Brooklyn Heights has Montague Street and DUMBO has a small, good cluster around Water and Front. Both neighborhoods go quiet by around 11pm on a weeknight. If you want a late night, you're getting on a train either way.
Space and price. Brooklyn is generally cheaper per night than Manhattan, and that gap is the strongest argument for crossing the river. But apartment-style stays exist on both sides now, and if what you want is a kitchen and separate bedrooms rather than a standard hotel room, compare like for like before you assume Brooklyn is the better value.
Getting between them. They're about a 30-minute walk apart over the Brooklyn Bridge, which starts a few minutes from our door. By subway it's one stop on the A or C from Fulton Street to High Street. There's also the NYC Ferry from Pier 11 at Wall Street across to DUMBO, which takes a few minutes and is the nicest commute in the city on a clear day.
When we'd tell you to stay in Brooklyn
If your trip is mostly Brooklyn: Prospect Park, Williamsburg, Smorgasburg, a show at BAM, friends in Carroll Gardens. If you're travelling with a photographer. If you've done the Manhattan sights on a previous trip and you want a slower week. If the nightly rate difference is what makes the trip work.
When we'd tell you to stay in the Financial District
If you're doing Midtown, museums, Broadway, or an office near Wall Street. If you're arriving from JFK or Newark and want the shortest transfer. If you're travelling with kids and the daily round trip is going to be the thing that breaks everyone. And if you want to walk to the 9/11 Memorial, Battery Park, the Statue of Liberty ferry and the Seaport rather than ride to them, all of which are 5 to 15 minutes on foot from John Street.
Where we fit
We're at 17 John Street, five minutes from Fulton Street station and a few minutes from the Brooklyn Bridge entrance. The suites are apartment-style with full kitchens and kitchenware, there's a laundry room, a fitness center and a guest lounge on site, and free wifi throughout. Check-in is self-service via a smart lock, so a late arrival from the airport isn't a problem, and we hold luggage in the lobby free of charge if you land before 4pm.
If you end up on the Brooklyn side instead, walk back over the bridge at dusk at least once. That's the one thing everyone agrees on.