Two of my children lived in Brooklyn, and we have several long-term friends there as well.
It's got some very liveable neighborhoods, nice neighbors, good restaurants. But the schools are lousy. My son had several kids in one of the few public schools with a good reputation, but a new principal came in and things quickly went downhill. One of my grandchildren received death threats from another student.
They pulled them out, sent them to a private school at great expense, and have now moved to Connecticut. Brooklyn is a great place to live if you like cities and if you have plenty of money for private schooling. The public schools are a disaster.
There are good schools in Brooklyn. I know several parents who have their kids in public school.
Hopefully, word gets out so that these hicks from Oklahoma and Ohio decide to stick with Manhattan, thereby making Brooklyn Brooklyn again instead of Manhattan East. The Yuppification of Brooklyn has gone too far.
Odd you should mention Connecticut. When I married my Hartford-born husband 32 yrs. ago, the first thing I learned when in Hartford was that I was expected to lock my car doors. I'm from eastern upstate NY and had never experienced that before. I moved back to where I grew up just so I didn't have to continue to live that way.
I don't know what the requirements are for homeschooling in the State of NY, but it seems the city would be GREAT for doing it because of the myriad opportunities for culture, history, etc. there. You could hop the subway and go on a field trip at least once a week!