Cross browser support is important for your website. It's a general usability issue; users should be able to access your content and should not be forced to use a particular web browser or technology to view your site. Adding to this your site should render consistently especially considering all the different web browsers options available now. Well that's all nice, but working as a web developer I know how challenging this can be.
As of late there has been a push to drop support for some older browsers, specially the IE6 No More initiative. And like I mentioned before your site should render consistently, but not exactly in different web browsers. I don't necessarily agree with dropping support for IE6 just yet; in particular if your website caters to an international audience (or wants to) or if your web analytics verify most of your users are viewing content with that browser.
I do agree that with older web browsers it's ok for you to limit some features of your site. If someone is using an older web browser it's fair to suggest their browsing experience will not be as rich. One note on this, the user should always be able to access the content. When I say limit features I mean that cool Ajax thing might not work, but the user can still navigate different pages they just have to reload the page.
As a developer most of my initial testing is done in the Firefox web browser. However, during development and pre launch you should test the following web browsers: IE 8-6, Firefox, Safari, Chrome, and Opera.