Are you using any password add-ons/programs?

Type about:preferences#security<enter> in the address bar. Look under Logins,

Note: For Firefox v56+ its; about:preferences#privacy Look under Forms & Passwords +++++++++++++++++++++++++++++++ Turn on Remember Logins. To check the passwords you have listed, press Saved Logins.

Press the Exceptions button and see what's inside.

I did that originally by following the instructions in Help section. That did not cause the password save screen to appear when I went to the web site and logged in.

Start Firefox in Safe Mode {web link}

A small dialog should appear. Click Start In Safe Mode (not Refresh). Is the problem still there?

Warning: There are some web pages that do not allow browsers to record or fill in the log-in information.

But if you go to the Mozilla Add-ons Web Page {web Link} (There’s a lot of good stuff here) you can try a Form Filling add-on. I found that the form-filler add-on I use is not affected.

So what you are saying is I should boot up in safe mode?

Did you verify that there isn't a login block exception stored for this domain ?

Also make sure that you do not run Firefox in permanent Private Browsing mode (Always use Private Browsing mode; Never Remember History).

I know it is not blocked because I was offered the choice to save logon and password before. I have never run Firefox in private mode.

If you click 'never' for saving the password then you create a block exception and you will have to remove this exception to make Firefox ask to store the password.

I want to thank everyone who took the time to help an old mn work this problem out. I am indebted to you all.