I was facing the same issue and I made it solved by installing one extension "Google Invisible reCaptcha"
Wait... I changed few modifications in JS code also.
After installing extension set Site key and secret in admin side and open subscription.phtml
YOUR-PROJECT/app/design/frontend/YOURTHEME/default/template/newsletter/subscribe.phtml
Add code just after Form Tag.
<div class="g-recaptcha" data-sitekey="YOUR-SITE-KEY"></div>
In the last of file add this JS Snippet.
<script src='https://www.google.com/recaptcha/api.js'></script>
<script type="text/javascript">
jQuery(document).ready(function(){
jQuery('#recaptcha_response_field').addClass('required-captcha-entry');
})
var yourFormValidationObj = new VarienForm('newsletter-validate-details');
Validation.add('required-captcha-entry', ' ', function(v) {
return !Validation.get('IsEmpty').test(v);
})
</script>
It worked for me. Cheers if it works for you..