You didn't tell us what you consider to be 'affordable', not did you tell us where you are willing to live. An affordable apartment in the Bronx is quite a bit different than an affordable apartment in Manhattan.
In order to educate yourself, take a look at the renting guide posted free by RDNY.com ( a great source of no fee listings, by the way). You can find the guide here: http://www.rent-direct.com/rentingguide/Welcome.html
Good luck with your search.
Paying advance rent and/or additional security is something some landlords will consider.
Bottom line is that no one can tell you anything with certainty without specifics. And even then, a broker can only guide you in the direction his experience takes him as it will be the landlord who has the final decision.