How do I see if Wi-Fi is connected on Android?

You should be able to use the ConnectivityManager to get the state of the Wi-Fi adapter. From there you can check if it is connected or even available.

ConnectivityManager connManager = (ConnectivityManager) getSystemService(Context.CONNECTIVITY_SERVICE);
NetworkInfo mWifi = connManager.getNetworkInfo(ConnectivityManager.TYPE_WIFI);

if (mWifi.isConnected()) {
    // Do whatever
}

NOTE: It should be noted (for us n00bies here) that you need to add

<uses-permission android:name="android.permission.ACCESS_NETWORK_STATE" />

to your

AndroidManifest.xml for this to work.

NOTE2: public NetworkInfo getNetworkInfo (int networkType) is now deprecated:

This method was deprecated in API level 23. This method does not support multiple connected networks of the same type. Use getAllNetworks() and getNetworkInfo(android.net.Network) instead.

NOTE3: public static final int TYPE_WIFI is now deprecated:

This constant was deprecated in API level 28. Applications should instead use NetworkCapabilities.hasTransport(int) or requestNetwork(NetworkRequest, NetworkCallback) to request an appropriate network. for supported transports.

Since the method NetworkInfo.isConnected() is now deprecated in API-23, here is a method which detects if the Wi-Fi adapter is on and also connected to an access point using WifiManager instead:

private boolean checkWifiOnAndConnected() {
    WifiManager wifiMgr = (WifiManager) getSystemService(Context.WIFI_SERVICE);

    if (wifiMgr.isWifiEnabled()) { // Wi-Fi adapter is ON

        WifiInfo wifiInfo = wifiMgr.getConnectionInfo();

        if( wifiInfo.getNetworkId() == -1 ){
            return false; // Not connected to an access point
        }
        return true; // Connected to an access point
    }
    else {
        return false; // Wi-Fi adapter is OFF
    }
}