Yes, I have many friends (10ish maybe) who aren't natives but currently live in Brooklyn. Queens is affordable too (you can buy a small house for ~$300,000ish IIRC, I think in Rockaway Beach) but you're very far away from Manhattan. I had another friend who lives in Long Island City (Queens, close to Manhattan) but her rent was high IIRC. I'm unfamiliar with the rent in The Bronx.
However NYC as a whole has high sales and income taxes though. Including City income tax.
Brooklyn is where you live when you move to The City without a lot of money - like for grad school.
Rent is (probably) higher in Park Slope or Williamsburg though which are more "desirable" areas of Brooklyn.
Living in a borough other than Manhattan is not the same as living "somewhere else", and 1800/month is certainly not the case if you consider all of NYC.
You could get cheaper, but you'd be far away from the actual city (at which point you might as well just live somewhere else).