Is there a plugin to protect downloads with recaptcha?

Is there any plugin to protect a download via recaptcha?

None that I know of… However, this could be achieved indirectly by using a download manager plugin such as this one which has the option to allow downloads only to logged in users. The indirect solution would then come in the form of using one of the multitude of plugins which require captcha/reCaptcha as part of the registration process for users on your site.