It seems that you are having some issues in order to set up your Slingbox PRO HD with an Airport Extreme router, since the connection drops.
First, we recommend you to verify if you are using a hard-wired connection between the Slingbox, the router and your computer. Make sure to verify that all the connections are properly set. If the Slingbox network light does not come up all of the time, we recommend you to test the connection using a different computer.
If the Slingbox light is off, I recommend you to check the following link and follow the included steps.
If the issue persists, you might be using a faulty power adapter. If this is the case, I recommend you to give us a call.