First you need to update your phtml ajax code like this.

    $(document).on('click','.import',function(){
                jQuery.ajax({
                url: '/admin/customimport/index/import',
                type: "POST",
                data: {customimport_id : '<?php echo $this->getRequest()->getParam('icustomimport_id'); ?>'},
                success: function(response){
                    console.log('ABCD');
                }
            });

Now replace your controller code like this.

    $id = $this->getRequest()->getParams('customimport_id'); 
    print_r($id);
    exit;

Or try this code.

    $postData = $this->getRequest()->getPost(); 
    $id = $postData['customimport_id'];
    print_r($id);
    exit;