Actually, I don't bother about it, since it shows me as logged out if I merely close the webpage without signing out...I'm using Chrome...and I used another computer to test it out...it only shows me as logged in when I actually open the site page...
Well, just in case, you need to check your log-in settings...do you allow your browser to remember the password to the site? If you allow it, then it may log in for you automatically...also, check the BD log-in window; do you click the 'remember me' check box? that may be another reason...
But like Slick said...it really is not a problem unless it's on a shared computer, for example...