I have lived in NYC for 12 years. Almost bought a place in Brooklyn last year. Are you saying Brooklyn is cheap?
If a household makes $250k/year and is responsible, then there are plenty of affordable places to live.
So, you can't live in Dumbo, Cobble Hill, or Brooklyn Heights (or parts of Williamsburg and Carroll Garden) - so what. There are plenty of amazing options with arguably the same amenities (schools, parks, housing stock, retail/dining, accessible transportation, etc.).
There is affordable real estate (which in chic Brooklyn <~$1 million) in Carroll Garden, Park Slope, Windsor Terrace, Prospect Park South/Ditmas Park, Greenpoint, and even sections of south Flatbush. I'm not sure if Prospect Heights or Williamsburg meet your safe critieria, but both of those sections have witnessed an influx of middle class families and capital.
In fact, most of "South Brooklyn" - Midwood and Bay Ridge being the two most accessible to Manhattan - are where most longtime middle class families live and have lived for generations. (but see below for a better conversation on this)
Safe doesn't always mean crime.
Then, what does it mean? As you can see above, I wouldn't call places like Williamsburg, Long Island City, Prospect Heights or even Red Hook safe even though they're well above middle class 'hoods in my book.
Some hoods are identity-safe, meaning if you're like everyone else they're probably really safe, such as Brighton Beach, Flushing, Harlem, Inwood, Richmond Hill, etc.), but even those places are probably safe across the board for incoming middle class families. Just as safe as, say, the East or West Village, Chinatown, Chelsea, Hell's Kitchen, etc. in Manhattan, which are actually higher crime hoods than most of the places I'll mention, if my memory serves me correct.
I'm sure the real estate next to the BQE, Atlantic and Flatbush avenue is affordable. But since you asked, Coney Island, Brighton Beach, Bay Ridge, Harlem, Jamaica, The Bronx, Bed Stuy, are not places you would want to raise a family in if you had a choice.
1. How is Bay Ridge unsafe?
2. Coney Island, Jamaica, and Bed Stuy are all more-or-less slums, but there are some notable exceptions of middle classness (Stuyvesant Heights in West Bed Stuy, Jamaica Heights near St. John's and Seagate in Coney Island). Brighton Beach is actually really safe, even more so if you're Russian or Eastern European (former Soviet).
Those are probably not great places for new-to-NY $250k earners to raise families, though, so you're right.
Bay Ridge does not deserve to be grouped into that list... I'm not sure if you've ever been, but it's about as nice as Brooklyn neighborhoods come.
Lots of nouveau riche young families have not only moved to Bay Ridge, as I've written above, but also to other south Brooklyn neighborhoods like Ditmas, Midwood, and Mill Basin.
I know you know this, but it should be said that there are lots of neighborhoods in the Bronx, a few of which have become havens for new middle class families who are priced out of Manhattan such as Riverdale, Parkchester [on Bronx Park], and Co-Op City. So, in fact middle class families who want space, good schools, and a safe community do occasionally choose to live in the Bronx, but I digress. Same goes for Harlem and Washington Heights. Lots of very nice middle class communities on the West Sides of both of those neighborhoods that have been strengthened via gentrification, etc.
In Queens, Sunnyside, Jackson Heights, Astoria, Forest Hills, Rego Park, and most of the Eastern Park of Queens county (from Flushing to Nassau Co.) are pretty awesome places to lead a middle class life. I'd include the Rockaway Peninsula (from the 80s over) in this, but that place got demolished during Sandy and likely won't be back for awhile, despite being inhabited by some of the more affluent communities in NYC. Furthermore, Southeast Queens (the JFK neighborhoods) are some of the most stable historical middle class black communities in the country.
Where do you live? (Also, where were you looking in BK). I'm in south "Victorian" Flatbush. Lots of new middle class families and money around these parts and it's been that way for a very long time.