Payment options do not display when https used only when http used
My hunch is that CiviCRM is looking for an AJAX callback with an absolute URL.  It'll go with what's in the Administer - System Settings - Resource URLs, which is in turn influenced by the base URL in civicrm.settings.php.  The solution is to have the same URL, with https, be in effect across your entire site.

For a number of reasons, including potential problems with your session ending when you follow an absolute URL to http from https or vice-versa, you should have your site use a single canonical URL, with everything else redirecting to it.  For example, you might decide that https://example.org is the proper URL.  You should set up redirects in Apache/Nginx to have http://example.org, https://www.example.org, and http://www.example.org all send a visitor to https://example.org.

Read more at https://civicrm.stackexchange.com/a/3233/44