Filter entries by ajax to controller
I did a little more research on Ajax and found a solution. I updated my Ajax such that the data line looks like:

$.ajax({
    method: 'get',
    url: 'actions/websiteListing/checkStatus',
    data: {filter: data},
    headers: {'Content-Type': 'application/x-www-form-urlencoded'}
})


Then in my controller, I was able to pass that data by adding $filter to the first line and including that for my 3rd $criteria line:

public function actionCheckStatus($filter)
{
    $criteria = craft()-&gt;elements-&gt;getCriteria(ElementType::Entry);
    $criteria-&gt;section = ('live');
    $criteria-&gt;hostSite = $filter;
    $criteria-&gt;limit = null;

    $entries = $criteria-&gt;find();

    foreach ($criteria as $entry) {
        echo $entry-&gt;title . '&lt;br&gt;';
    }
    die();
}