I did some quick searching on Hostelworld.com (which despite the name lists also B&B's and cheap hotels)...
Cheapest private rooms in Manhattan I could find were around USD 78 (39 per person). This would be somewhere such as La Sienna, which is nowhere near downtown Manhattan, but far up in Harlem.
So, if $50-60 (for two) is a hard limit, a hostel dorm is what you have to go with, and even then it won't be downtown. $35 per person will get you a bare-bones bunk in 6-bed dorm at Jazz on Amsterdam Avenue in Upper West Side, for example. For a dorm in a better hostel closer to downtown, you'd have to pay upward of $49 per person.
So, like others said, what you want (decent room for two downtown for $50-60) doesn't seem possible.
As for other options, besides getting lucky with Airbnb or such, you can of course try e.g. Couchsurfing or GlobalFreeloaders and stay at someone's home for free, but there's no guarantee you'd find a host (especially if you are a new user with no references).
Btw, I stumbled upon a place called Urban Oasis (in Midtown) where a basic double private room costs $110; looks nice & has great reviews. In your shoes I'd probably go with something like this.