You can activate the ReCaptcha feature in merchant panel at Configuration > Features.
After activation go to Configuration -> Captcha, and enable google recaptcha there. You have to set your private (secret) and public (site) keys. You can get them from https://www.google.com/recaptcha/admin/create
It is very important to set the keys correctly, otherwise ReCaptcha will not show up in your affiliate signup form.
ReCaptcha is not applied when affiliate is added using PAP API, or using Affiliate manager in merchant panel.
You can choose one of the visual themes: