Today I've got one interesting question from my co-worker that I've never think of (because I always have this option turned off in my browser). How to force browser running APEX application to prompt to save password on login page? With a little googling about HTML from elements in a few seconds I've got answer. It's pretty simple. The only thing you have to do is to set page property "Form Auto Complete" (under Security region of page properties) of login page to "On".
If you have turned on right properties in your browser it should work as a charm (even in IE).
There are some security issues regarding turning this option on, but this is not topic of this post.
Personally, I don’t use this option and probably never will. :)