Google recently announced changes to their mobile search algorithm, which now favors "mobile friendly sites" in search listings. All other things being equal, websites that are optimized for mobile devices get preference in mobile searches.
Many of our clients have received scary spam email in the last few weeks along the lines of "we looked at your site and it isn't mobile friendly, Google is going to remove you from their listings!" Hogwash.
Having a mobile-friendly or responsive website is important, especially if you do any email marketing. People read email on their phones, and that's where they click links. Your board members probably view your website on tablets. At the other end of the spectrum, a lot of folks using desktop computers now have very large HD displays, and your site may look like it's floating off to the side of their browser window. Responsive is the only way to build a new website.
It's also costly. It takes at least three times as long to design and build a responsive website, which means it costs at least three times as much. In some cases, an existing website design can be modified to become more or less responsive, but even that can take a significant amount of time. Simply put, every change we make must now be tested at six or more different sizes.
Also, many of the values in a responsive website's stylesheet are now relative to each other, rather than fixed values, because they have to change with the screen size. The code is harder to write, and changes to one element can have implications for other elements, which makes testing at all those different screen sizes that much more important.
So, yes, the spam is at least partially true: you do want a responsive website. And if you're concerned about being competitive in mobile search results with other sites that do what you do, then, yes, you need to upgrade your site. But if not, don't worry about it for now. But be prepared: your next website is probably going to cost more than your current one did, because responsive sites are not as easy to design and build.