You ready for this? It *USED* to work. I've got the same issue ever since we upgraded our DNN to 04.08.03--I am no longer prompted to save/remember passwords in both IE and FireFox. My best guess as far as I can tell is the core got modified to restrict this, using the autocomplete="off" method. If I remember correctly, versions older than 04.08.02 did not implement this.
FYI, there's a dupe discussion going on about this here.
A quick Google Search will give you plenty of folks that have documented this, but it is very clearly outlined at the Mozilla Developer Center here. Most of the time you'll see financial institutions utilizing this to increase security for their end user's accounts.
IMO, from the developer/admin standpoint, this is a great security enhancement; but for the end user this makes life ridiculously complicated. I've got 60-some-odd DNN portals that each have a unique password. I fondly remember the day where my browser would remember all those passwords for me.
Here's two solutions I'd say you can do.
1.) The easy, but cumbersome, Way Out. Store all your passwords in encrypted text files or an encrypted directory on your local machine/network storage. Sure, you'll have to either remember them all or do a lot of copying and pasting—but for some/many, this is easier than #2.
2.) Modify the Core. Have fun with this one =) Yeah that's not much help, but hey...I haven't had time to look deeper into that yet. If I do come up with a solution, it'll be posted.