This can be done using WP_Query using author and tax_query.
Something like this:
$args = array(
'post_type' => 'post',
'author' => get_current_user_id(),
'tax_query' => array(
array(
'taxonomy' => 'custom-taxonomy',
'operator' => 'EXISTS'
),
),
);
$query = new WP_Query( $args );
And then check if posts are returned through this query.
Please note that this code is not tried or tested and may contain syntax errors.