A lot of people are getting here by searching for things like "putty inactivity", so I thought I'd answer your question for you. You'll kick yourself at how easy it is.

After a period of time, PuTTY likes to kick your connection when you're not doing anything. It's mostly so that if you're away from the computer, someone else won't come and start playing with your box.

Start PuTTY, go to the 'connection' tab, and change the 'keepalive' value to something like 120. Don't go much lower than that, or your server might get sick of null packets.

Whilst you're in your options, you may want to increase the security of your connection so no one steals access to your server. If you have PuTTY open a lot, you might be interested in being able to send it to the system tray, instead of having it open in a new window.