To disable commenting in your journal, go to http://www.insanejournal.com/editinfo.bml and uncheck the option labeled "Enable message boards," which is found near the bottom of the page, then click "Save Changes".
You also have the option to disable comments on individual entries, while leaving commenting enabled for the rest of your journal. Using the web client, you can do this from the full version of the Update Journal page at http://www.insanejournal.com/update.bml?mode=full . While posting or editing the entry, check the box marked "Disallow Comments" to disable comments on that entry only. If you are disabling comments on an entry that already has comments, these comments will then become invisible.
Many clients available for download include the ability to disable commenting on individual entries. For more information on your preferred client, consult its documentation or its menu options.