This is a question I see raised many times during consultations where a client has already spoken to some amateur web design agencies before our own meeting takes place and have been told that having a responsive website is going to cost them extra!
First off this is nonsense, responsive design is industry standard and it has been since April 2015 so a web designer or web design agency shouldn't be even offering to build you a non responsive website in the first place, secondly, they should not be charging you more as all websites built after 2014 should be responsive and work on a range of devices otherwise you are missing out on a massive potion of the market.
Statistics show that over 59% of initial searches are carried out via mobile devices and this is set to rise even higher in 2016 so having a non-compliant website will definitely lose you business making having a responsive website is no longer an option it's now a necessity and we have been using responsive code for well over two years in all our web design projects even budget sites so in 2016 there really is no excuse.
For some agencies and designers its simply and excuse to hike up the price for many it is simple a lack of understanding of responsive code especially the amateur design agencies who are only able to use templates or drag and drop designer software, for these types of companies it's a way to put you off so that they can continue to use their outdated website building software.
Another thing that you should be aware of is the fact that web design is a totally unregulated industry which attracts lots of rogues and amateurs who not only churn our sub standard template websites created by open source drag and drop software but also misuse words such as responsive and bespoke cause because they have heard them being use by other to sell a higher quality service.
A professional web design company will build you a responsive website as standard and over the coming year many won't even make a big deal about it...