Unlike most Argentine hotels, US hotels vary their prices widely depending on bookings, attempting to maximize revenue by maintaining a full house at the best possible prices. It's not uncommon to find the same room available one day for $150 and $400 the next. Also, the better places sometimes discount more heavily on sites catering to foreign visitors (like Despegar, for example) than to their domestic customers.
I'm from NYC, but I no longer maintain a place there. If I'm planning a longer stay - something over one week - I may look for a furnished apartment. Most owners won't rent these for just a few days, however, and in my experience, the cheap rental flat will have the same problems - or worse - as the cheap hotel. Plus no way to argue for an upgrade or change of room if it turns out to be truly terrible.
For shorter stays, I just check the major booking sites - hotels.com, despegar.com.ar, etc. - and sort by price. Except for the most prime tourist seasons when there's also a huge convention in town, there are always good buys available.
I usually make a booking that can be cancelled without a charge, and then check back a few days or a week before I'm going to see if something better and/or cheaper has appeared. A hotel with unexpected vacancies may offer amazing last-minute rates. On my last short trip up, I paid $300/night for a room with a rack rate of $890. This isn't cheap, but it was good value. I also found OK hotels in good locations for $125-$150.
There are also specialized sites in the US that track the deepest discounts.
http://www.travelzoo.com/hotels/new-york/ today highlights 3 deals at perfectly acceptable hotels for $139 x night. These almost always involve non-refundable prepayment, so make sure it's what you want before you buy it.
When you find an affordable place, if you'd like an ex-New Yorker's opinion, send me a PM.