mysql> CREATE TABLE test (id INT UNSIGNED NOT NULL AUTO_INCREMENT PRIMARY KEY, dt DATE);
Query OK, 0 rows affected (0.41 sec)

mysql> DELIMITER @@;

mysql> CREATE TRIGGER tr_bi
    -> BEFORE INSERT ON test
    -> FOR EACH ROW
    -> BEGIN
    ->     IF new.dt < '2001-01-01' OR new.dt >= '2020-01-01' THEN
    ->         SIGNAL SQLSTATE '45000' SET MESSAGE_TEXT = 'Wrong date entered.';
    ->     END IF;
    -> END;
    -> @@;
Query OK, 0 rows affected (0.11 sec)

mysql> DELIMITER ;

mysql> INSERT INTO test (dt) VALUES ('2010-01-01');
Query OK, 1 row affected (0.05 sec)

mysql> INSERT INTO test (dt) VALUES ('2030-01-01');
ERROR 1644 (45000): Wrong date entered.

mysql> INSERT INTO test (dt) VALUES ('2000-01-01');
ERROR 1644 (45000): Wrong date entered.

mysql> SELECT * FROM test;
+----+------------+
| id | dt         |
+----+------------+
|  1 | 2010-01-01 |
+----+------------+
1 row in set (0.00 sec)

mysql>