MainActivity.cs

using System;

using Android.App;
using Android.Content;
using Android.Widget;
using Android.OS;
using Android.Net;
using System.Threading.Tasks;

namespace NetworkCheck
{
	[Activity (Label = "NetworkCheck", MainLauncher = true, Icon = "@drawable/icon")]
	public class MainActivity : Activity
	{

		protected override void OnCreate (Bundle bundle)
		{
			base.OnCreate (bundle);

			// Set our view from the "main" layout resource
			SetContentView (Resource.Layout.Main);

			// Get our button from the layout resource,
			// and attach an event to it
			Button button = FindViewById<Button> (Resource.Id.myButton);
			
			button.Click += async delegate {
				var connected = await IsConnectedToNetwork();
				Console.WriteLine ("Connected: {0}", connected);
			};
		}

		Task<bool> IsConnectedToNetwork ()
		{
			var tsc = new TaskCompletionSource<bool> ();

			Task.Factory.StartNew (() => {
				var connectivityManager = (ConnectivityManager)GetSystemService (Context.ConnectivityService);
				var wifiState = connectivityManager.GetNetworkInfo (ConnectivityType.Wifi).GetState ();
				var internetState = connectivityManager.GetNetworkInfo (ConnectivityType.Mobile).GetState ();

				bool connected = (wifiState == NetworkInfo.State.Connected || internetState == NetworkInfo.State.Connected);

				tsc.TrySetResult (connected);
			});

			return tsc.Task;
		}
	}
}