Unfortunately, the cookie is not the problem here. By default the cookie expires as soon as the browser AND all other instances are closed, and even if it hasn't, its absolutely no good if the session expires. (If you want to set the lifetime of this specific cookie yourself, use the session.cookie_lifetime value and set to something other than 0, see http://www.php.net/manual/en/ref.session.php.)
If a client has cookies enabled, comes to the site, gets a session cookie, closes the browser AND all instances of that browser, and comes back to the site, the user has to get a new cookie. Easy.
The culprit is the SID, that little session ID that tags itself to the end of relative URLs if the cookie cant be set. That thing leads PHP to the appropriate session file (vincent?) as I understand it. If this is on the tail of a URL that's bookmarked and someone else enters using this bookmark while that session is still active (user A gets up from a community computer, user B sits down and finds the bookmark and follows it), there is no log in required (if thats what you did to allow the session to be set). In any rate, the new user has the control of that session. If this is a secured site, i.e., credit card usage, sensitive company information, etc., this is a BIG problem. As Vincent said, turn SID off! Tell the user No Cookie, No Service, sorry. Unless, of course, you feel like taking the heat for the outcome that is.
Jim