The specific answer to this particular question is:
Once you have set up reCAPTCHA, you should see a small icon on the bottom right-hand side of the page where it's installed. That will only appear if reCAPTCHA has been set up correctly and is working. The only other way to tell is to log into your Google reCAPTCHA dashboard after your website has received a considerable amount of traffic. Once this has happened, the dashboard will be able to display statistics. But these only appear after the page has received sufficient traffic. Before that, the only way to tell is by the icon appearing at the bottom-right of the page where reCAPTCHA is installed.