how to append a css class to an element by javascript?

var element = document.getElementById(element_id);
element.className += " " + newClassName;

Voilà. This will work on pretty much every browser ever. The leading space is important, because the className property treats the css classes like a single string, which ought to match the class attribute on HTML elements (where multiple classes must be separated by spaces).

Incidentally, you're going to be better off using a Javascript library like prototype or jQuery, which have methods to do this, as well as functions that can first check if an element already has a class assigned.

In prototype, for instance:

// Prototype automatically checks that the element doesn't already have the class
$(element_id).addClassName(newClassName);

See how much nicer that is?!

Adding class using element's classList property:

element.classList.add('my-class-name');

Removing:

element.classList.remove('my-class-name');