While my MacBook sleeps, the network drops off?
There are a couple of things that could be causing this:
Make sure "Wake for network access" is checked in Energy Saver. This should work if Skype is sending data over Bonjour.
Your prior computer may only be turning off the display instead of sleeping the computer entirely. In the Energy Saver control panel, try setting the computer sleep time to Never. You can retain your display sleep time.
My understanding is that #2 is likely the solution as I don't believe the type of traffic on which Skype relies will trigger the Wake-On-LAN feature.
Have you tried disabling standby mode?
sudo pmset -a standby 0
Some newer laptops enter a lower energy mode after about an hour of normal sleep under certain conditions. I don't know if it also disables waking for network access though.