Solution

Application.internetReachability is what you need. In conjunction with Ping, probably.

using UnityEngine;

public class InternetChecker : MonoBehaviour
{
    private const bool allowCarrierDataNetwork = false;
    private const string pingAddress = "8.8.8.8"; // Google Public DNS server
    private const float waitingTime = 2.0f;

    private Ping ping;
    private float pingStartTime;

    public void Start()
    {
        bool internetPossiblyAvailable;
        switch (Application.internetReachability)
        {
            case NetworkReachability.ReachableViaLocalAreaNetwork:
                internetPossiblyAvailable = true;
                break;
            case NetworkReachability.ReachableViaCarrierDataNetwork:
                internetPossiblyAvailable = allowCarrierDataNetwork;
                break;
            default:
                internetPossiblyAvailable = false;
                break;
        }
        if (!internetPossiblyAvailable)
        {
            InternetIsNotAvailable();
            return;
        }
        ping = new Ping(pingAddress);
        pingStartTime = Time.time;
    }

    public void Update()
    {
        if (ping != null)
        {
            bool stopCheck = true;
            if (ping.isDone)
                InternetAvailable();
            else if (Time.time - pingStartTime < waitingTime)
                stopCheck = false;
            else
                InternetIsNotAvailable();
            if (stopCheck)
                ping = null;
        }
    }

    private void InternetIsNotAvailable()
    {
        Debug.Log("No Internet :(");
    }

    private void InternetAvailable()
    {
        Debug.Log("Internet is available! ;)");
    }
}

 

Things to note

  1. Unity’s ping doesn’t do any domain name resolutions, i.e. it only accepts IP addresses. So, if some player has Internet access but has some DNS problems, the method will say that he has Internet.
  2. It’s only a ping check. Don’t expect it to be 100% accurate. In some rare cases it will provide you with false information.