If you are having issues were the Captcha label shows in a form but reCaptcha does not appear. it is because of a change that occurred when Google took over reCaptcha. The location of the API changed from api.recaptcha.net to www.google.com/recaptcha/api. Modifying the original code to reflect what is seen below in bold should fix the issue. Only change the url for the RECAPTCHA_API_SERVER.
Note: I used "HTTPS" instead of "HTTP" this may not be necessary and it may work fine with just "Http" but i didn't test it.
edit this file:
Original Code:
const RECAPTCHA_API_SERVER = "http://api.recaptcha.net";


class plgCaptchaRecaptcha extends JPlugin
const RECAPTCHA_API_SERVER = "https://www.google.com/recaptcha/api"; //MODIFIED code
const RECAPTCHA_API_SECURE_SERVER = "https://www.google.com/recaptcha/api";
const RECAPTCHA_VERIFY_SERVER = "api-verify.recaptcha.net";