If you're only targeting modern browsers:

Use element.classList.add to add a class:

element.classList.add("my-class");

And element.classList.remove to remove a class:

element.classList.remove("my-class");

If you need to support Internet Explorer 9 or lower:

Add a space plus the name of your new class to the className property of the element. First, put an id on the element so you can easily get a reference.

<div id="div1" class="someclass">
    <img ... id="image1" name="image1" />
</div>

Then

var d = document.getElementById("div1");
d.className += " otherclass";

Note the space before otherclass. It's important to include the space otherwise it compromises existing classes that come before it in the class list.

See also element.className on MDN.