Either go for ...
SELECT *
FROM employees
WHERE Children NOT LIKE 'E%';
if you want all records where the child's name doesn't start with E
or for ...
SELECT *
FROM employees e1
WHERE NOT EXISTS (SELECT 1
FROM employees e2
WHERE e1.Employee = e2.Employee
AND Children LIKE 'E%');
if none of the returned employees should have a child that starts with E
.
Check out version one and version two.
p.s. based on your structure it's
SELECT E.EMPLOYEE_NAME,
E.EMPLOYEE_TITLE,
E.EMPLOYEE_AGE,
C.CHILDREN_NAME
FROM EMPLOYEE E INNER JOIN CHILDREN C ON E.EMPLOYEE_ID = C.EMPLOYEE_ID
WHERE NOT EXISTS (SELECT 1
FROM CHILDREN C2
WHERE E.EMPLOYEE_ID = C2.EMPLOYEE_ID
AND C2.CHILDREN_NAME LIKE 'E%');
Check this Fiddle.