Alternatively, one could make it opt-out rather than opt-in. Slashdot does this for your session - you can mark a session as being at a "public terminal", ensuring that you're logged out when your browser session ends.
Slashdot, news.yc, reddit, fark or digg aren't exactly the most important of my accounts. I sure wouldn't want my bank to leave me logged in by default, but I find the rest of my accounts to be near worthless.
My problem with the "remember me" checkbox is that I don't want the site to remember my login credentials until after I know that my login attempt was successful. If I mistyped my email address or password (the latter being both easier to mistype and harder (impossible) to verify), I don't want the site to remember that information.
Fortunately, my browser usually asks me whether it should remember the login information after the page is loaded.
The "remember me" checkboxes don't store login information, they just setup a permanent cookie rather than a session cookie (so the cookie — and login — survives restarts).
Remembering credentials is a feature of the browser, some will let the page load through (letting you check that the credentials are correct) while others (safari does that I believe, and it's extremely annoying) block the whole UI and force you to hope you didn't make a typo, or they're going to store the wrong one (though they'll ask again when you enter the right one, and will replace the incorrect one. Still annoying)
That is a different thing. That is a browser function to store your credentials and be able to auto-fill them later. Opera does it nicely by displaying a bar on top of the screen after you submitted. So you can actually see if you logged in and then decide to store and not to store.
This submission is about server-side (cookies or likewise) though.
