What minor version of SmarterMail 16.x are you using? I don't recall any bugs relating to this functionality in earlier versions and can confirm that in the latest release, 16.0.6362, the browser will remember the user's last choice. On new installations, for new accounts, and if the browser cache is cleared, the left side panel defaults to open. It should only close if the user clicks on the button with the hamburger icon in the upper left corner.
If you're running an older version, try an update to the latest. If the issue persists, submit a ticket to our Support Department so we can look into this more.