"Remember me" is always a hole in a security, whatever you use - cookie or session - someone can (in theory) steal a cookie, and thus enter account without any password. There are ways to increase security, by allowing recalling only under same IP. Cookies are AFAIK less secure however that depends on way you implement auth. For instance, by keeping password hash in cookies you almost give intruder a real password (which is a no-no, it may be used on other webapps) as there are dictionaries to obtain simple passwords from hash. Salting hash helps not much.
All in all, session is the simplest and well enough way. Use tables if you use more than one webserver for an app, otherwise sessions on disk is ok.