Hi Theo.  If this is your first time to New York, I'd advise you to stay in Manhattan, simply because it is safe and easy to get around using public transportation. Brooklyn is great, but traveling there can be difficult - everything is more spread out. I can't recommend a specific hostel, as I've never stayed in one in NYC, but glancing at hostelbookers.com, Hotel 309 might be a good fit for you. It's in an interesting neighborhood (the Meatpacking District, near Chelsea and the West Village) and it's close to both the A and the L train, which will take you to downtown Brooklyn and Williamsburg, respectively. Their 4-person dorms start at $25 a night, which is a better bargain than most Brooklyn hostels listed.

sometimes you can find awesome deals here- https://www.airbnb.com/

You can get short term apt rentals, sublets, or room rentals. 

You could try  Sheraton Brooklyn New York Hotelor  Aloft New York Brooklyn they are both just 3 stops away from Manhattan