You may want to try enabling TCP Keepalives under the Connection menu for PuTTy. Start by loading your profile, navigating to the Connection menu on the left side, and under Low-level TCP, click on "Enable TCP Keepalives."
You'll also need to adjust the first option, the "Sending of null packets to keep session active."
Try changing that to 10, 20, or 30 (which is in seconds).
For the changes to stick, you'll need to save your profile once these changes have been made, else you'll have to do this before you attempt to login each time.
