For add new class in select ...you need to flow below steps:
Step:copy app\code\core\Mage\Checkout\Block\Onepage\Abstract.php
to app\code\local\Mage\Checkout\Block\Onepage\Abstract.php
Here you have find the function getCountryHtmlSelect($type)
COde
$select = $this->getLayout()->createBlock('core/html_select')
->setName($type.'[country_id]')
->setId($type.':country_id')
->setTitle(Mage::helper('checkout')->__('Country'))
->setClass('validate-select')
->setValue($countryId)
Change to and i have added a new class mynewclass
..
if($type=='billing')
{
$select = $this->getLayout()->createBlock('core/html_select')
->setName($type.'[country_id]')
->setId($type.':country_id')
->setTitle(Mage::helper('checkout')->__('Country'))
->setClass('validate-select mynewclass')
->setValue($countryId)
}else{
$select = $this->getLayout()->createBlock('core/html_select')
->setName($type.'[country_id]')
->setId($type.':country_id')
->setTitle(Mage::helper('checkout')->__('Country'))
->setClass('validate-select')
->setValue($countryId)
}