Filter Table Before Applying Left Join

You could also do:

SELECT c.Customer, c.State, e.Entry
FROM Customer AS c
LEFT JOIN (SELECT * FROM Entry WHERE Category='D') AS e
ON c.Customer=e.Customer

SQL Fiddle here

You need to move the WHERE filter to the JOIN condition:

SELECT c.Customer, c.State, e.Entry
FROM Customer c
LEFT JOIN Entry e
   ON c.Customer=e.Customer
   AND e.Category='D'

See SQL Fiddle with Demo