How can I make reCAPTCHA a required field?

if you want to use the native html5 popups, than here is the solution

JavaScript:

window.onload = function() {
    var $recaptcha = document.querySelector('#g-recaptcha-response');

    if($recaptcha) {
        $recaptcha.setAttribute("required", "required");
    }
};

CSS:

#g-recaptcha-response {
    display: block !important;
    position: absolute;
    margin: -78px 0 0 0 !important;
    width: 302px !important;
    height: 76px !important;
    z-index: -999999;
    opacity: 0;
}

I had the same problem as yours and solved it this way:

First declare a variable that stores 1 or 0 depending or whether the user filled the capcha correctly.

var allowSubmit = false;

Then you need a function which gets executed when the user fills the reCapcha correctly:

function capcha_filled () {
    allowSubmit = true;
}

... and a function that gets executed when the reCapcha session expires:

function capcha_expired () {
    allowSubmit = false;
}

To tell reCapcha about your functions (callbacks), set those data-attributes in your html:

<div class="g-recaptcha"
   data-callback="capcha_filled"
   data-expired-callback="capcha_expired"
   data-sitekey="your site key"></div>

Or if you use explicit load:

  var onloadCallback = function() {
    grecaptcha.render('your_div_id', {
      'sitekey' : 'your_site_key',
      'callback': capcha_filled,
      'expired-callback': capcha_expired,
    });
  };

You need also a callback for the form submission:

function check_if_capcha_is_filled (e) {
    if(allowSubmit) return true;
    e.preventDefault();
    alert('Fill in the capcha!');
}

Finally add in the form the onsubmit attribute:

<form action="..." onsubmit="check_if_capcha_is_filled">

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Note: as mentioned in the comments, a server validation is still needed. The code prevents accidentally submitting the form unless the capcha is filled and is only for user's convenience.