The button becomes active on our forgot login after the user has supplied the requested details. Typically Username and/or Email Address. You can see this working on our demo site below.
demo.cbdemosites.com/community-builder/forgot-login
If it is not behaving like the demo site for you then you likely have something causing a fatal JS error and breaking its jQuery. Press F12 to open your browsers developer console and select the Console tab then refresh the page and see if any fatal errors (they'll be red) exist.