I've followed all of the responses to this query.  While I have our main company file in QBD, we have added a second business on the same QBD.  When I log into that business, it DO NOT ASK for a password.  I'm not sure why this is so different.  However, I have followed the instructions for the staying logged in for 90 days, change it to 90 days, saved the option, and closed QBD.  Opened up QBD again and it's still asking me for a password.  

Similar to a lot of the commenters, I am the only person who uses QBD in this business and the office is in a family home.  There is NO security risk whatsoever. 

Edited to add, you need to go to FILE, CLOSE COMPANY.  This is when it takes effect.  Simply clicking the X at the top right of QBD doesn't do anything.