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 → Energy Saver → 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.