Best resources for finding an apartment in Civic Center, New York City?

I would either go to Craigslist under the Downtown and Financial District headings or directly to a broker's website. Citihabitats and Bond are the biggest real estate brokerages and they have all of their vacancies listed. But, a broker will charge you 10% of a year's rent to put you in an apartment so try the "no fee" and "by owner" listings on Craigslist first. It's, honestly, pretty doubtful that you'll be able to find a place without a broker because the vacancy percentage in New York is so small right now; but, hey, it's worth a shot.