Hello mannumbapos,
When you sign in to your QBO account, you only have the option to remember your user ID.
The password is saved on your browser. The steps to save it may differ on every browser.
For Google Chrome
For Internet Explorer
Though your browser prompts to save your password automatically, you can also do it manually. Just go to your browser settings. You can ask your IT personnel to help you with it.
Please let me know if you have more questions about this.