How are you persisting the user's login information? In a cookie or in session?
If it's in the cookie, then I'm not sure what the problem could be. The cookie will be sent to any browser in the web farm so that shouldn't cause a problem.
If you are using session then what type of session management are you using? If you are using In-Proc than it is likely that your users are getting bumped from one server to another and thus their session data is not there on the other server. You can use SQL bases session state to eliminate this problem.