Can I use AirDrop if one or more of the participating computers is asleep/off?
It should work as long as you have Wake for network access enabled in  System Preferences &#x2192; Energy Saver &#x2192; Battery/Power Adapter.

Depending on the capabilities of your Mac, the option's description may differ:


  
  Wake for network access - Your Mac supports Wake on Demand over both Ethernet and AirPort
  Wake for Ethernet network access - Your Mac supports Wake on Demand over Ethernet only
  Wake for AirPort network access - Your Mac supports Wake on Demand over AirPort only
  
  [source]
  


The option "wake for Ethernet network access" is mentioned only to round up the answer. Wake on demand for ethernet will not help if your Macs communicate using Wi-Fi.

However, you can enable AirDrop to communicate using Ethernet as well as explained here. This will also enable AirDrop on so-called unsupported Macs.