Don't know if this is what you have in mind but you don't rule it out in your OP: you can also use craigslist to sublet the apartment of a New Yorker who is going to be away during your dates. I live in NY and sublet my apartment regularly in this way. Yes, you have to use common sense while using craiglist - make contact with the letter, get as much information as you can, get references from past subletters if possible, just be sensible and prudent.
I disagree with the poster above who rules out above 96th and outside Manhattan - parts of Brooklyn especially are very close to Manhattan and are extremely pleasant (Park Slope, Fort Greene, Williamsburg) and you'll get lower rates there. Above 96th on the east side I don't know well but on the west side that is largely Columbia University housing and up to 120th or so you'd be fine. Depending on your age group, yes, Alphabet City might be a bit noisy and gritty but then again it has changed hugely in recent years and I find it a very pleasant place to be now.
There are also house swap or apartment swap websites which a friend has used with success in NY. And websites such as (I may be misquoting this) sabbaticals.com, which academic friends have used, and I'm sure it's open to non-academics too!
Good luck.