Is this area a good homebase for sightseeing in New York? To be more concrete: How long do I need by public transport to the major sights? How much does this cost? Is it a safe area in New York?
This zip code finder shows 11217 as the neighborhood of Boerum Hill.
This is a very nice part of Brooklyn. You have a choice of several nearby subway lines (Atlantic Ave - Barclay's Center is a major transit hub), which will get you into downtown Manhattan in 10-15 minutes, midtown in maybe 15-20 more. Smith Street has a lot of very good restaurants, as do 4th and 5th Aves. The area should be quite safe.
[Edit from a Brooklyner: Boerum Hill is a very low-crime area; you can look up crime statistics on NYC.gov for any neighborhood. Here's Boerum Hill's. It's still part of the city, so be smart about not displaying valuables, etc., but it's a very nice neighborhood.]
I'm not sure what you mean by "how much does this cost". If you mean the transport, as of 2017, New York City has a flat fare of $2.75 for all subway trips, including transfers, and local buses. (You can also buy various kinds of passes, see http://www.mta.info/ for details as well as transport maps, etc.) If you mean living in general, I'd expect Boerum Hill to be less than Manhattan but still fairly expensive. For lodging, I think you'll just have to see what hotels are charging. For food, there is a wide range of choices from cheap pizza at $2-3 per slice to gourmet restaurants charging $30-50 and up for dinner.
I would call it an excellent sightseeing base if it is in your price range.
The Boerum Hill neighborhood is not a bad option. I have never been there at night; however, it is a mostly residential area and I at least have never felt unsafe. While the media has made much of some recent incidents of violent crime, similar incidents happen in any gentrifying neighborhood, including the adjacent neighborhoods of Fort Greene or Park Slope.
It is a good location for transportation connections into Manhattan. For quick access to Midtown Manhattan there is a Long Island Rail Road (LIRR) terminal at Atlantic Avenue and for the City as a whole as there are many subway lines near the intersection of Atlantic and Flatbush Avenues, namely
- 2 and 3 Trains for the Financial District, west side of Lower Manhattan, Greenwich Village, Broadway/Theatre District
- 4 and 5 Trains for Battery Park and everywhere on the East Side from Union Square to Grand Central to all the museums on the Upper East Side
Atlantic Av-Pacific St
- D Train for Midtown and the Upper West Side; outbound to Coney Island
- N Train for the east side of Lower Manhattan, and Queens
- R Train for South Ferry, Lower Manhattan, and Queens
Atlantic Av (a different one from above; check the signs at the entrance)
- B Train for Midtown and the Upper West Side, or in the opposite direction to Brighton Beach
- Q Train for the east side of Lower Manhattan, and Queens
A little farther north you can get to
- Hoyt - Schermerhorn
- A and C Trains for Greenwich Village and the Upper West Side
- G Train if you have a lot of time to waste
On the western side of the neighborhood you have access to
- Bergen St
- F Train for Midtown Manhattan, Roosevelt Island, and Queens
Being in Brooklyn, it will take you some time to get uptown. The 4 Train from Atlantic Av to 86th Street for (e.g. for the Guggenheim Museum) would take at least 45 minutes, plus walking time. On evenings and weekends there is often construction and trains may be rerouted, and if you must transfer lines you will also need to account for headways of 20 minutes or more during off-peak times. Plan your days using sites like HopStop or 511 NY and remember that part of the fun of Manhattan is walking its streets, especially for distances under about 2½ miles (4km) or so.
Fares on the New York City Subway system and MTA buses are a flat $2.25. If you purchase a Metrocard with at $10 of value, you get a 7% bonus. If you will be spending a few days in New York there is a 7-day unlimited ride pass you can purchase from the same vending machine for $29.