Add dropdown arrow indicators to menu items that have submenus only?

No. CSS has no contains child selector. You'd probably be better to just add a class to the li element. For example:

<li class="has-child">
    <a href="#">The Link</a>
    <ul class="child">
        <li>Child 1</li>
    </ul>
</li>

Your CSS selector would in turn look like:

.mainNav li.has-child > a:after {
   color: #444;
   content: ' ▾';
}

You could have jQuery add the class for you, if that's an option:

$('.mainNav li:has(ul)').addClass('has-child');

jsFiddle Demo

CSS has no contains child selector. However it has various sibling selectors, only-child and not(:only-child) Since you add indicator to the anchor, use following CSS

 .mainNav li>a:not(:only-child):after {
       color: #444;
       content: ' ▾';
    }

<div class="mainNav">
    <li>
        <a href="#">The item with child</a>
        <ul class="child">
            <li>Child 1</li>
        </ul>
    </li>
    <li>
        <a href="#">No child item</a>
    </li>
</div>