Have you ever dealt with a situation when the client reports a bug, but you could not reproduce it? Even though you are maybe using the same browsing environment and following the same steps, you still cannot reproduce the same problem. In some instances a simple clearing of the cache may be the quickest solution to fix the problem.
I encountered one browser related issue recently. The client had reported a bug that appeared after updating his system to OS X Maverick. Numbers were hidden partly by the input box. This was because the height and width values were not specified. The default value was verified when the same browser was used.
Therefore, testing among different browsers is important. It is the most efficient method to get rid of such bugs. Different default size may hide important data as well as buttons after that. I found this particular bug in IE, where the pop up window for file uploading had a limited width. The long input box pushes the in-line upload button outside of window and the overflow is hidden.
But in chrome and safari everything is correct because the input box is not as long. I will list some related issues I found: for example some plugins can disable the advertising; some browsers have javascript disabled; some do not support HTML 5; some do not have flash enabled. It is very important to let the client know which browsers and operating systems are supported so that they don’t try to reproduce problems outside of this set.
What about a small group of people using outdated browsers like IE6? It largely depends on the specific requirements. If you want people using IE6 to get the same experience as others, then such effort may entail lot of work to ensure backward compatibility.
Otherwise, it may be in the best interest of the user to update their browser to enjoy the full benefits of a website they are browsing.