How important is a responsive website design to rank highly in Google?

A good design is always the key to success. If you site is responsive it will open up nicely in all the platforms like tablet, smartphone, desktop and laptop.

It is not mandatory to have a responsive website to boost your overall rankings, however it is recommended as it will serve the following purpose:
1. There will be more time spent on your site that will boost your rankings.
2. Bounce rate will reduce, that will boost your rankings
3. Google will list it in a better way, that will boost your rankings.
4. It will be visible in proper formats from different platforms that will boost your rankings because more people will open it.
5. Search engines will fetch the links more efficiently, that will boost your rankings.

I hope that helps, for more information feel free to setup a call.

Short answer: Super important.

Here is why:
1. Google explicitly penalizes non-mobile enables websites
2. People have a poor experience on non-mobile enabled websites from mobile phones, which further kick-in other google algorithms to lower your website.

Example: One of the signals google uses to rank a page is the time spent on a website. If someone using a mobile phone comes to your website from Google and jump off it because it doesn't render well. The time spent on website will be low, and google will lower your rank.
3. Mobile internet browsing has officially overtaken desktop browsing - http://bgr.com/2016/11/02/internet-usage-desktop-vs-mobile/

I would highly recommend not ignoring mobile users when thinking of building a website.

Hope this helps.

Hello, I know you have received some responses here already but I feel like there were some wrong thoughts provided here.
I have been helping startups and Small/Medium companies grow their traffic through inbound marketing. At my small hybrid marketing agency, we stay up to date on Google SEO requirements (Google releases a document every so often with changes) <-- we follow this document.

One of the previous answers mentions that you don't need to be mobile friendly. This is wrong, you do need to be mobile friendly and more so you need to design a website with the mobile first approach. If you want to achieve first-page ranking for a particular phrase by paying someone to give you a ton of thin/fake backlinks then you might not need a mobile friendly website but the results won't last. Probably won't last more than a month.
Now you must consider that is not about having a 'responsive' website more so than it is about the code you use to build it, the tags you use, and the information you provide to sites like Google to let them know you have an optimized mobile experience including being responsive.

With that said, nowadays is almost ludicrous to have a website that isn't mobile-friendly at a minimum because it will hurt your conversion rates. So even if is not for SEO purposes, you do need to have a mobile friendly website.

Starting (best guess) around 2016 Q3-Q4 Google has begun deindexing desktop views of sites. Meaning, they're removing desktop site views from all search results.

If you're SEOing your site, then having a solid pass (100% pass) from Google's mobile testing tool is essential.

So the answer to your question is yes, if you're targeting SEO - having your site show up in natural search results.