OP, your title is misleading, because from it alone it looks like you're telling to never save login information in the browser, but in your post you are only talking about public computers. There's nothing wrong with saving logins on your own computer, and if you use a master password, it's nearly as good as a full password manager, because your passwords will be encrypted.
I Agree
Firefox has an in built password manager, lockwise.
If you auto save your logins in firefox this won't happen. But you should never do this is on public computers anyway.
Additionally, every service states clearly not to save your logins in public computers.