I've lived in Brooklyn, Manhattan, and Staten Island and I'd have to go with Manhattan.
Crown Heights Brooklyn is about as authentic of an NYC experience as you're going to get (I lived one block away from Utica Ave) but Brooklyn in general is too big, and it's a hassle to leave the neighborhood when you have to go somewhere. Also, the nightlife ain't really nothing past midnight. Brooklyn has a lot of creative types though. A lot of people that you meet there are lowkey really skilled.
Staten Island is obviously very isolated and doesn't have much to do outside of family life and drugs. There's some cool dudes over there, but it's really just hood near the ferry and suburban everywhere else. It's great for having a base away from the city where you can just chill and be next to the water. Also, the panorama view from Victory Blvd is the best in NYC.
Manhattan wins because it's NYC all of the time. You never have to leave the island because everything is within walking distance and there's always something going on. This is ideal because you save hours of your day from having to commute to any of the outer boroughs and can get things popping right after work. Plus the night doesn't end, it just transforms into the morning, which is cool because you don't have to go anywhere. Another thing that Manhattan has going for it are the kinds of people that you meet. While Brooklyn has the cool kids, Manhattan has the rich kids. And that tradeoff is worth it if you want to keep pace with your ambitions. The city life in general is good towards that end.