When you design a website these days, make sure to do it responsive. You already heard of this before, I guess. In the article The Best Browser is the One You Have with You you can find a lot of arguments why it's so important.
More and more people access the Internet via a mobile device. As Luke Wroblewski points out, already 50% of Facebook's users access the web from their mobile device. And most of them use their browser instead of the native app to do so. The variety of devices goes from feature phones with a limited browser over smartphones to desktops with a browser capable of the latest web technologies.
It's impossible to know which device and especially which browser the user will use. And because of the wide variety of devices with different specs, your site should adapt to the possibilities on that device. You cannot expect that the user has to adapt his device to use your site. Who liked it to open a site optimized for 1024x768px with a smaller monitor ten years ago?
Especially should your site open the same Url on different devices and serve the same content. Users might start browsing your site on their desktop browser, leave house and continue browsing on the go. Firefox offers to open your tabs from other browsing sessions on other devices, for example.
Even when mobile users are still rare on your website, start building it future ready today. The move from desktop to mobile won't stop.