A Google Webmaster Help thread asks if there is an SEO benefit to going the cookieless domain route.
The simple answer is that no, there is no direct SEO impact on going with the cookieless domain. But it can impact site speed, which is now a direct SEO factor, although it likely won't have a serious impact if any on your rankings if you do use cookies, since most of the web pages on the internet that you visit and rank high in Google do have cookies.
Google's John Mueller explains why people use cookieless domain:
The idea behind a cookieless domain is that it can speed up transferring static files by the browser not having to send cookies with every request. This is something that can help to reduce the time needed to render a page, if you have static, shared content.
But does it impact rankings? John said no:
A cookieless domain does not have any direct SEO effect. However, as mentioned there are situations where we may choose to use site speed as a ranking factor, so if you have a site with extremely slow-loading pages, it may make sense to review this as well as the other page-speed recommendations to help speed things up.
For more on cookieless domains, see this page.
Forum discussion at Google Webmaster Help.
