Either this doesn't work, or it doesn't provide enough feedback on how to run it properly. This is what I get when running this command:

s@s-Air:~$ banshee --stop-when-finished

Unhandled Exception: System.ArgumentNullException: Cannot handle a null message; maybe the bus was disconnected
Parameter name: msg
  at NDesk.DBus.Connection.HandleMessage (NDesk.DBus.Message msg) [0x00000] in <filename unknown>:0 
  at NDesk.DBus.PendingCall.get_Reply () [0x00000] in <filename unknown>:0 
  at NDesk.DBus.Connection.SendWithReplyAndBlock (NDesk.DBus.Message msg) [0x00000] in <filename unknown>:0 
  at NDesk.DBus.BusObject.SendMethodCall (System.String iface, System.String member, System.String inSigStr, NDesk.DBus.MessageWriter writer, System.Type retType, System.Exception& exception) [0x00000] in <filename unknown>:0 
  at Banshee.PlaybackController.IPlaybackControllerServiceProxy.set_StopWhenFinished (Boolean ) [0x00000] in <filename unknown>:0 
  at Halie.Client.HandlePlayerCommands () [0x00000] in <filename unknown>:0 
  at Halie.Client.Main () [0x00000] in <filename unknown>:0 
  at (wrapper managed-to-native) System.AppDomain:ExecuteAssembly (System.Reflection.Assembly,string[])
  at System.AppDomain.ExecuteAssemblyInternal (System.Reflection.Assembly a, System.String[] args) [0x00000] in <filename unknown>:0 
  at System.AppDomain.ExecuteAssembly (System.String assemblyFile, System.Security.Policy.Evidence assemblySecurity, System.String[] args) [0x00000] in <filename unknown>:0 
  at (wrapper remoting-invoke-with-check) System.AppDomain:ExecuteAssembly (string,System.Security.Policy.Evidence,string[])
  at System.AppDomain.ExecuteAssembly (System.String assemblyFile) [0x00000] in <filename unknown>:0 
  at (wrapper remoting-invoke-with-check) System.AppDomain:ExecuteAssembly (string)
  at Booter.Booter.BootClient (System.String clientName) [0x00000] in <filename unknown>:0 
  at Booter.Booter.Main () [0x00000] in <filename unknown>:0

The man page says that it should take either a 'true' or 'false' value, so I have tried variations such as;
banshee --stop-when-finished true
banshee --stop-when-finished=true
banshee --stop-when-finished='true'
And so on, all of these produce the same error message.