For registered users who used to use this feature, until I get a chance to implement it in this new design (which may be at least a couple of weeks), there is a workaround. Login normally, go to your profile page (using the "My profile" link at the top right), then logout from that page. You get presented with a login page there, which has the "remember me" option.

OK, the new login bar is there now with the "Remember me" feature.