I have some old sites that are old, and look it, just because I never found they urgently needed updating. For sites like amazon.com and craigslist they usually want the site to work for the lowest common denominator, including dial up and old browsers, so they avoid the bells and whistles. Awhile ago when I was working as a product manager in a silicon valley startup, a large percentage of the support calls we got were from users who had lower end hardware, OS, etc. so you had to decide whether to support them or just punt (i.e. tell them to upgrade their system). Notice how microsoft has never punted and even many dos programs can still run on windows 7. Apple on the other hand, regularly dumps its old users and says, your old programs will not work on our new stuff.
But I think looking modern and up to date can get you more business (assuming of course that you aren't the industry standard already, like some of the aforementioned sites)