I figured it out with quite a bit of help from Matt Gibson Answering my own question, just in case someone stumbles on this post trying to solve a similar issue.
You can add a class to a menu item. Go to Screen Options tab of the Menus page, select the CSS Class checkbox, and add the class to the desired menu.
Curiously, classes are added to top level menu items in WordPress differently from how they’re added to the lower level items. At the top level, classes are added to the <a>
tag, whereas at the lower levels (submenu, sub-submenu etc.) they are added to the
<li>
tag. A bit illogical, but that’s how it is.
If you want to make all the submenus a certain width (for example, 400px) you have to add the same class to each of them, and then write this in your css file:
#access ul ul li.someclass > a {
width:400px;
}
On my Twenty Ten, things were already put in with a sub-menu class, so without adding the Screen Options, I was able to add the following to my style.css file:
#access ul ul.sub-menu {
background-image: url('some-backing-image.png');
}