Limit loop to future events

Assuming your date is in the correct format for proper ordering (yyyy-mm-dd), add a meta_query to your args:

$args = array( 
    'post_type' => array('basic_events'),
    'order' => 'ASC',
    'orderby' => 'meta_value_num',
    'meta_key' => 'date_of_event',
    'paged' => $paged,
    'meta_query' => array(
        array(
                'key' => 'date_of_event',
                'value' => date( 'Y-m-d', current_time( 'timestamp' ) ),
                'compare' => '>',
                'type' => 'DATE'
        )
    )
);

EDIT– Changed date( 'Y-m-d' ) to date( 'Y-m-d', current_time( 'timestamp' ) ) to get the date relative to your site’s timezone settings. See current_time in Codex for info.