My recent experience:
My former supervisor claimed, in earnest, that because Google Analytics told us that "87% of our users have Java enabled and less than 3% are using IE6," that we didn't have to worry about supporting older browsers or users with JavaScript disabled.
Problem 1: Java is not JavaScript.
Problem 2: In order for Google Analytics to track a hit, the browser must have JavaScript enabled because the GA interface is a JS include. GA is not, and can not, be aware of users with JS disabled, which can potentially severely skew its reports.
Problem 3: one of our biggest customers requires that all engineers use IE6 with JS disabled.
Problem 4: The boss (at the time) didn't know how to read analytics reports.
If you want to know how important this support is to your business, a good place to start is the IIS logs. Just about everything related to the browser caps is stored by IIS. I regularly import the logs into SQL Server and run some basic reports from my admin site, which come in handy every time someone starts suggesting that we go crazy with the jQuery BS.
If you decide to start building complicated, script-dependent interfaces, be sure that your interface degrades gracefully and doesn't remove required functionality if JS is disabled.