Common Javascript File for all Themes
Yes, you can use the Libraries API for this.  The docs explain all of this fairly well.   This is best for things that change infrequently.

For my custom Javascript libraries (which I work on continuously), I make module with an instance of hook_library

function mymodule_util_library ()
{
    $path = drupal_get_path("module", "mymodule_util");

    $libraries = array();

    $libraries["mymodule_util"] = array(
        "title" =&gt; "MyModule Javascript Utilities", 
        "version" =&gt; "0.0",
        "js" =&gt; array(
            $path . '/UTIL.js' =&gt; array(),
            $path . '/UTIL.HTML5.js' =&gt; array(),
            $path . '/UTIL.COLOR.js' =&gt; array(),
            $path . '/UTIL.MATH.js' =&gt; array(),
        ),

        'weight' =&gt; -10,
    );

    return $libraries;
}


Put the module in "site/all/modules" so all of your sites can see it, then enable it on the sites you need.

Then, wherever you need these JS libraries, you use drupal_add_library

drupal_add_library("mymodule_util", "mymodule_util", $every_page);


Typically, this is in a hook_block_view or a preprocess function.