How do I output a database option that is an array into a get_posts array?

You could use wp_parse_id_list() to sanitize your comma seperated list of post IDs. It’s defined as:

function wp_parse_id_list( $list ) {
    if ( !is_array($list) )
        $list = preg_split('/[\s,]+/', $list);

    return array_unique(array_map('absint', $list));
}

where we can see that it returns unique array values as well.

Note that absint() is defined in core as abs( intval() ) and there’s a difference in the max output of intval() for 32 and 64 bit systems according to the PHP docs.