The best I've been able to come up with is the following. It might not be a very clean solution but you might find it to be suitable (or it might at least point you in the right direction). It uses several Juicebox-Pro API methods and events and also jQuery which you will need to include in your web page.
When the gallery is first loaded (onInitComplete), if Small Screen Mode is detected (getScreenMode), the .jb-area-caption container is removed from the Document Object Model (DOM).
This works fine but will only be of use if the Splash Page is not used. As soon as the gallery is expanded (from the Splash Page or from the Expand Button), the .jb-area-caption container gets reinstated but removing the container as before as soon as onExpand is fired does not work due to a timing issue. This is why I have introduced a small delay before removing the .jb-area-caption container again.

<!--START JUICEBOX EMBED-->
<script src="jquery-1.11.1.min.js"></script>
<script src="jbcore/juicebox.js"></script>
<script>
    var jb = new juicebox({
        containerId: 'juicebox-container',
    });
    jb.onInitComplete = function() {
        if (jb.getScreenMode() === 'SMALL') {
            $('.jb-area-caption').remove();
            jb.onExpand = function(expanded) {
                if (expanded) {
                    setTimeout(function() { $('.jb-area-caption').remove(); }, 500);
                }
            };
        }
    };
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

I hope this helps.

I just found this and it worked when putting it in the CSS for smaller screens:

.jb-caption p {
    display: none;
}

However, this only works in the app when not expanded. Now I need a solution for when it is expanded.

If should work for any value of screenMode. The code should be executed only if screenMode="SMALL" or if screenMode="AUTO" and Juicebox chooses to use Small Screen Mode.

The code works when the gallery is expanded in the same page. However, on iOS devices, the gallery is expanded in a new page (the 'full.html' file within the 'jbcore' folder). Please see the Expand Gallery Behavior support section for more details.
Try the following (in addition to the code above in your main embedding web page).
Open your gallery's 'jbcore/full.html' file in a plain text editor and change:

<!--START JUICEBOX EMBED-->
<script>
    var expanded_jb_gallery = true;
    new juicebox({
        containerid:'juicebox-container',
        galleryHeight:'100%',
        galleryWidth:'100%'
    });
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

... to:

<!--START JUICEBOX EMBED-->
<script>
    var expanded_jb_gallery = true;
    var jb = new juicebox({
        containerid:'juicebox-container',
        galleryHeight:'100%',
        galleryWidth:'100%'
    });
    jb.onInitComplete = function() {
        if (jb.getScreenMode() === 'SMALL') {
            $('.jb-area-caption').remove();
        }
    };
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

(Further testing today suggests that you may also need to increase the timeout from 500ms to 1000ms in my original code for when the gallery is expanded in the same page.)