I was having the same issue. It turns out that I had 2 reCAPTCHA plugins installed which were conflicting with each other. This is what I had installed
* Contact Form 7
* Better WordPress reCAPTCHA
* Contact Form 7 reCAPTCHA extension
I deactivated Contact Form 7 reCAPTCHA extension and then the BWP reCAPTCHA tag generator worked. Not sure if this will help your issue, but you only need these two plugins installed:
* Contact Form 7
* Better WordPress reCAPTCHA
