Save web site usernames but not passwords in Firefox

Found here:

Firefox will remember passwords by default. You may have disabled this feature, or told Firefox to never remember passwords for a particular site.

1. At the top of the Firefox windowOn the menu bar, click on the ToolsFirefoxEdit menu, and select Options...Preferences....
2. Select the Security panel.
3. If it isn't already check marked, check Remember passwords for sites.
4. To the right of Remember passwords for sites, click the Exceptions... button .
5. Make sure that the site you're trying to log in to isn't in the list. * If it is, select the entry and click Remove.
6. Click Close to close the Exceptions window.
7. Click OK to close the Options windowClick Close to close the Preferences windowClose the Preferences window

You should look at the form saver add-on "Lazarus: Form Recovery".

By default it doesn't save passwords, but it saves all the other fields in the form:

You should look at the add-on lastpass.

This keeps all your passwords online and encrypted so only you can access them. However, they're accessible anywhere in the world.
It has too much functionality to detail here, but is quite complete regarding features.
Available for Windows, Mac and Linux, in my experience it's the best password manager.