I don't believe that any of us are developing applications for browsers which don't support cookies. The problem is users who have disabled cookies because they feel they are a security problem or don't want to be tracked.
Cookies are required for hotmail, gmail, ect. and are a standard feature of all of the major browsers today, and have been since version 2 or 3 (I think).
The question to ask yourself as a developer is if it is worth it to put in the work needed to use your site without cookies to satisfy the small portion of users who will have them turned off.
The conclusion i've reached at this point is that if you are attempting to create a secure login process you are going to use cookies combined with ip address checking, which will help mitigate man in the middle attacks, and subsequent users on the machine.
--Will