Network monitor is your best application ever to know if you can access the Internet or not. It doesn't care if you're on WiFi, protected WiFi (such as hotels, planes and coffee shops) or corporate network. All that matters is that your computer can reach the Internet and that you know whether or not it's working!
Finally, if you are concerned about privacy regarding this application, the Privacy page has more detailed information about what Network Monitor discloses from your computer. The short version is: your IP address (and nothing else) to a single server (220.127.116.11).