When my Captivate project is published to my LMS, the course opens a new window. You are able to copy the URL of this window, and access the course without logging into the LMS. How can I prevent this?

Go back to your LMS administrator and ask them why they haven't set it up more securely. The scenario you have mentioned should not occur if the LMS is properly secured.

Some LMSs have the course content reside in a folder OUTSIDE the web space so that users cannot access the course URLs directly.  For example, Moodle does it this way and then pulls the course content into a temp folder when the learner clicks the link to request a module.  Whatever way your LMS is supposed to do it, it's not currently set up properly.