The best way to get NY Times is to use a site downloader like plucker or a commercial version (which I like much better) such as Offline Explorer and then convert it with Isilo. You can get ANY part of the paper (or the whole thing) and filter out the crap you dont want. It also supports passwords and login in scripts(and much, much more).
As for the question, I used to have the problem where sometimes it would ask for the password and not d/l anything but the login screen. It turned out that on the NY Times theres a logout button on each page. Download that link and next time it wants u to login in again. It was a real problem. Thankfully, Offline Explorer (as I am sure the other same-type progs all do) allows you to filter out links. I filtered out
http://www.nytimes.com/logout and never had the problem again.