Usually I try to accommodate organizations who tell me to go elsewhere by going elsewhere for my news, but this is, after all, the New York Times, and just by living in New York you wind up getting sent a link here or a restaurant review there or a Mark Bittman recipe there, or just something that sounds vaguely interesting, and suddenly you're at the site over 20 times for a month without having had any chance to judge whether each random link you've seen is worth using up one of your chits, and you see this screen.
You're supposed to think, "Well, this is obviously a valuable resource and I should totally give my money to this industry, even though I pay for no other news site on the web. This is utterly indispensable!" And then you fill out the form and give them your credit card number and go on your merry way, poorer, but wiser.
Instead, what you can do is get by the paywall through one mouseclick, two keystrokes, a delete key, and the enter key. You don't need to leave the site, you don't need to cheat, bootleg, or otherwise inconvenience yourself beyond the requisite five clicks mentioned in this paragraph.
Go up to the address bar. Click on it. Find the end of the actual link, which will end in ".html" and be followed by a ?. That's the actual link, and everything beyond it is the cruft that gives you the paywall warning. So delete the ? and everything after. Hit enter again.
Read the article. Know that everything that came after the ? cost the NY Times $40 million. If they'd kept the money, they'd be able to afford to keep their articles free for a lot longer.