Currently when you open MapBrowser Classic the following settings are saved:
Base map selected
Street and property overlays
If you find that when you open the MapBrowser these settings have reset, then you are most likely using the New MapBrowser. The feature allowing the New MapBrowser to save these settings is being developed and will be released in the coming months.
Another cause could be that your browser is set to delete cookies upon closing. You can fix this by going to your browser's settings menu and changing your cookies so that they are not automatically deleted. Please see your browser's help for appropriate instructions.