As far as "peddling influence" goes the liberals use the Times as their "truth" - their "blog" if you will. What you have, in effect, is a group of like-minded people reading like-minded journalism.
What are their circulation numbers?
The last report I saw was 1.5 million with most of that on the east coast, and a small portion spread across the country. For the print media 1.5 million isn't a lot of papers, and they exist, as all papers do, because of their advertisers who cater to the Times liberal market.
Is the NY Post gaining on them like Fox has exploded?
I haven't figures on the NY Post, but they have been a thorn in the side of the Times for many years, and is an ideological alternate to the Times in the NY area. The Post has taken advantage of the Times most recent credibility problems, and in this respect I'd have to "assume" that its readship has grown just as Fox grew because of CNN's credibility problems.