No. In fact it is harmful.
Technically, you can't expire a session cookie-- the moment you set an expiration date, it becomes a persistent cookie.
I think what you mean to ask is "Is it best to store a session ID in a session cookie or in a persistent cookie?" I would recommend a session cookie, as it is held only in memory and is removed from the cookie jar the moment the user closes the browser. If you use a persistent cookie, it's stored in the filesystem, and it isn't dropped until it expires or until the user clicks the logout button-- and sometimes they forget. The persistent cookie will stick around a while, meaning that a malicious user could come along a few moments later, open a browser, and resume the user's session.