Mavericks has a lot of nifty features, and because Apple thinks everything they do is nifty these features are turned on by default. A co-worker of mine recently upgraded to Mavericks on his work machine (I won't do that until 10.9.1 is released next week), and he discovered that every time he walked away from his computer he lost most of his SSH sessions. He did some testing, and found that only SSH connections in SecureCRT were dropping, while test connections in Terminal were not.
Long story short, since I don't want to blow up Gene's ego in regards to his work diagnosing this, the problem turned out to be App Nap. App Nap is a new feature that puts applications into a suspended state while the screen saver is active, thereby reducing power consumption.
Why wasn't Terminal affected? Apple engineers are well aware that sleeping a terminal app will cause dropped connections, so they completely removed the App Nap option from Terminal. You can't turn it on if you wanted to. All other apps default to enabled.
So, if you are having issues with dropped connections, or other application wonkiness when your screen saver kicks in, head on over to here for the fix: