Allow me to share a little information about the verification process when logging in to QuickBooks Online.
Verifying your identity helps you protect your data and your Intuit account. This is to make sure that you, and only you, can access your account. However, if you wish to remember the system your user ID and password, you can check your browser settings to prevent on clearing your history's data after closing the browser.
Open the Chrome Toolbar (3 lines to the right of the URL window) and then select Settings.
Scroll down and click Advanced.
Go to Content settings.
Disable Keep local data only until you quit your browser.
I've attached screenshots to guide you through the steps.
As of now, the remember my log-in credentials in QuickBooks Desktop (QBDT) is currently not possible. I understand how this option can benefit you and other users when logging in into there QBDT software.
I'll submit this feature request on your behalf to our product engineers so they'll consider adding this in the future.
To learn what's new and what's coming in QuickBooks, you can visit our Product Updates page for reference.