Apache goes to sleep on new OS X 10.9
I think the answer is "Stop pressing the Power Button".

I had read that tapping the power button in Mavericks would turn off the monitor. In fact this is wrong, it puts the machine to sleep. 

It is a very light sleep because the machine will still respond to 'ssh' requests, and the moment one tries to ssh into the machine the httpd will magically wake up. But, if no one tries to ssh, the httpd daemon will stop responding a minute or so after one taps the power button.

I was sometimes tapping the power button to turn of the monitor. I have stopped doing that. So far so good. 

More updates if this turns out to be the wrong answer.