If you design a website or write content for for it, you are entitled to all the benefits of the intellectual property. Although the vast majority of webmasters never become involved in copyright disputes, sometimes website owners ask if they should file a copyright for their work.
In short, no. There is no reason why you should file for a copyright. In the United States and most industrialized nations, all intellectual property is automatically copyrighted from the data of creation. So as long as you have proof that your content was created on a certain date– Google Search’s cache can often provide this– you’re good to go. Filing a copyright for something as quickly-changing as a website would be a needless hassle and expensive.