micahs question doesn't match his title- I don't believe it is the element's class name but the actual class he wants to modify.
In IE and firefox you can manipulate the document.styleSheets object and add, remove or edit the class declarations, but many browsers don't support this, and IE and firefox don't do it in the same way. It is the closest approach to what you want to do, however.
The easiest way (that works in the most browsers) to modify a class with runtime defined styles is to redeclare the class and put the css text for the declaration in a new style element, appended to the head of the document.
This works in most browsers. You might use it to do a rewrite of one or several classes, to set up a user stylesheet, say, but you don't want to be using this technique often. It adds some overhead, and can cause the page to flicker, depending on the changes you define.
While you can't call oldclass.newclass to ammend a class declaration,
you can add any class already defined to the element's className attribute.
Use element.className='oldclass newclass';
or element.className+=' newclass'; ( with a leading space in the string)
Anything from oldclass that newclass does not redefine will still be in effect.
Because while the class property is apparently a text string, browsers split the string on spaces, and apply each individual class, in the order it appeared in the string.