I'm not aware of any direct API that accesses the status of Airplane Mode directly but essentially it shuts down network availability so you could test for that using the DeviceNetworkInformation class. (It's a good idea to test this on a device but I believe this would simulate Airplane Mode)
public bool IsAirplaneMode()
{
bool[] networks = new bool[4] { DeviceNetworkInformation.IsNetworkAvailable, DeviceNetworkInformation.IsCellularDataEnabled, DeviceNetworkInformation.IsCellularDataRoamingEnabled, DeviceNetworkInformation.IsWiFiEnabled };
return (networks.Count(n => n) < 1);
}
If you'd like to ask the user to turn it on or off, you can do that by How to set Airplane mode in Windows Phone using C#