Back when I developed my first website in 1998, the idea of responsiveness was no where on my radar. Mobile devices as we know today were not available and the standard dimensions of monitors I believe only went to around 1024x768 pixels, at least in general use. So web sites were typically designed for a set resolution. These were the days of Netscape 4 and IE 4/5.
So much has advanced since those days and the idea of a web site either not working or not formatting correctly on a mobile device should be unheard of. Yet, here we are with some sites still not working correctly or offloading to a "mobile only" version. While some design still for a mobile responsive site, the goal should be for a mobile first / fallback mode to be the default, with things scaling for larger displays. This is the direction I take ALL design work here at I Do Web Work. Building up from mobile first can allow for a site, which if stylesheets were disabled, to simply work. Additionally any reflow of content for the viewer should not break the nature flow of the site's content with regards to how search engines or assistive technology sees your site.
Find out why it is so important to address a mobile first web when you work with me on designing or redesigning your next project.