As a web designer it is your duty to enlighten your client the advantages of redesigning their website to suit the mobile phone traffic.
In this case not many will agree if their current website is working fine and finding enough visitors for their business. Under the circumstances it would be better for you to focus on the other aspects of the website because the site might be experiencing problems with the visual designs required for the current web trend.
The other aspect is the content management (CMS) – as seen on this Wedding Photography website – that is not in vibe with the current marketing trends or the absence of a tool capable of updating the content. Focusing primarily on these two aspects would hugely benefit the client’s site and the client’s budget. Some sites would not match with the current trend of marketing the company is conducting on the street and customers generated by these means would find the website grossly disappointing when they visit it.
A website that is in union with the physical marketing conducted by the company (like the CityCard Manchester one is) would always look updated and attract more business. The redesign would not only improve the aesthetic aspects of the website but also enhance user experience. It would be important to integrate the new design with a CMS that would also take care of the other requirement of updating the website.
When you put this proposal before the client you would also mention that you would make the site responsive as you build it. The usual question you will encounter at such times from the client is whether it would add to the cost and your answer should be “Yes” but you should prelude it with a conversation highlighting the advantages of a responsive design. The point is that you should be able to convince the client who has already declined to spend money on a redesign suitable for mobile phone users