Based on what I saw in the MT support forum, the most common problem is the site URL not being in the same domain as the CGIPath URL. So, if your “Remember Me” option isn’t working, verify that the URL to your CGI is correct.
Another reason it may not work is if there are different directories involved. If the user is in one directory when making comments and another one to view the comments and post more, the cookie has to be set twice because it stores the path in the cookie. It’s more convenient if the directory that is used to post comments is the same directory that is used to retrieve them, but not strictly necessary, but it will require users to store their values twice.
While experimenting with this feature, Mozilla’s cookie manager was an extremely useful resource, allowing you to view and remove cookies stored in the browser.