Google now allows to hide the Badge, from the FAQ :

I'd like to hide the reCAPTCHA badge. What is allowed?

You are allowed to hide the badge as long as you include the reCAPTCHA branding visibly in the user flow. Please include the following text:

This site is protected by reCAPTCHA and the Google
    <a href="https://policies.google.com/privacy">Privacy Policy</a> and
    <a href="https://policies.google.com/terms">Terms of Service</a> apply.

For example:

So you can simply hide it using the following CSS :

.grecaptcha-badge { 
    visibility: hidden;
}

Do not use display: none; as it appears to disable the spam checking (thanks @Zade)

I have tested all approaches and:

WARNING: display: none DISABLES the spam checking!

visibility: hidden and opacity: 0 do NOT disable the spam checking.

Code to use:

.grecaptcha-badge { 
    visibility: hidden;
}

When you hide the badge icon, Google wants you to reference their service on your form by adding this:

<small>This site is protected by reCAPTCHA and the Google 
    <a href="https://policies.google.com/privacy">Privacy Policy</a> and
    <a href="https://policies.google.com/terms">Terms of Service</a> apply.
</small>

Set the data-badge attribute to inline

<button type="submit" data-sitekey="your_site_key" data-callback="onSubmit" data-badge="inline" />

And add the following CSS

.grecaptcha-badge {
    display: none;
}