The answer is a definite YES!!! The technology keeps changing with the touch screens gets popular, tablets getting smaller and smart phones getting bigger you never know what your audience using to when they visit your site. Instead of designing a site for different screen sizes you can target all devices with one responsive website.
Before the responsive website people had different versions of website and based on the device used the website would appear. The problem with this was that it had to make too many assumptions on the screen size and the device plus you had to compromise on the contents. With the responsive sites you just prioritize and stack them in order. The same set of contents will appear on the site and people would just have to scroll through it.
Top reasons why you should have a responsive design:
Increasing mobile usage
Usage of mobile phones is increasing day by day. People are using mobile while doing other task so it has become a norm. If you are into statistics it’s 20% of the people who are using mobile while other things. Judging from the picture and Einstein’s theory I know you can relate to the picture.
One site many devices
This is the major reason for responsive. You don’t have to worry about which device people are using they get the same contents. Thank god for media queries which made it all possible. Instead of making assumption of the device it can simply scan for screen resolution and fit the website in that resolution.
Search Engine friendly
It has become mandatory for sites to go mobile as search engines rank these sites higher. 20% of the Google searches are on mobile and people access social media sites mainly from mobile than desktops so you can understand the importance of having a responsive site.
Easier to manage
With the responsive site you need to design just one site. Ofcourse best practice is to start from small devices and then move on to big ones. The site becomes more modular and easily adapted to the environment it is in. With this sort of design it is easy to manage the data and you can choose what to show first and highlight the necessary.