This article assumes that you know what responsive design is. If you don't, head on over and read "Responsive Design".
Leave your Print Expectations at the Door
It's unreasonable for us to expect that a webpage will always render pixel-perfect. Our job as designers is to mitigate these differences and get them down to a level that we can deem acceptable. Browsers all have their individual quirks are strengths; if we choose which quirks to iron out wisely, your users won't notice and you'll save time, money and headaches. Many of the fixes that are employed to deal with these browser's short-comings slow down your site and create additional bugs, so they're best to avoid.
All Things Considered
It's important to note that "responsive design" is a far departure from what we've seen in the past. Unfortunately, we can't take your existing site and "make it responsive". Responsive design isn't a feature or plugin: it's a whole new way to think about Web design. Not only will the "shell" of the website need to change, you also have to be aware that your content pages may also have to be tweaked individually. This problem will be compounded if you resorted to using tables for layout or have pages heavy on images/graphics.
The Best Approach is Fresh Approach
I know a lot of us are in love with our trusty website. Countless hours in front of a computer tweaking content, sometimes late into the evening, making things "just so". Don't get me wrong, concentrating on the details is great, but every now and again it pays to step back and look at the big picture.
Have a look at your analytics. What's performing well? Talk to your clients. What did they like about your Web presence? What prompted them to call you? How many people are viewing your website on phones? tablets? How are these trends changing?
Responsive Design is a game changer, but it's still early in the game. If you have the time and the spirit, a move right now could help give you an advantage over you competition. On the other hand, If responsive design is a part of a new strategy, you might blow them out of the water.