LEFT JOIN conversion to INNER JOIN, can't change FROM clause
That one caused me quite a bit of headache before I found a halfway elegant solution:

WITH x AS (
    SELECT BASE.id AS b_id
          ,BASE.start_date AS b_start_date
          ,BASE.end_date AS b_end_date
          ,CHILD.id AS c_id
          ,CHILD.start_date AS c_start_date
          ,CHILD.end_date AS c_end_date
          ,CASE WHEN DATE '2011-1-1' BETWEEN CHILD.start_date AND CHILD.end_date
                                     OR CHILD.id IS NULL THEN 0 ELSE 1 END AS c_hit
    FROM   BASE LEFT OUTER JOIN CHILD ON (BASE.ID=CHILD.ID)
    WHERE  BASE.ID = 1
    AND    DATE '2011-1-1' BETWEEN BASE.start_date AND BASE.end_date
    )
SELECT b_id, b_start_date, b_end_date
      ,c_id, c_start_date, c_end_date
FROM   x
WHERE  x.c_hit = 0

UNION ALL
SELECT b_id, b_start_date, b_end_date
      ,NULL, NULL,         NULL
FROM   x
GROUP  BY b_id, b_start_date, b_end_date
HAVING min(c_hit) = 1;


Of course, the better solution would be to get rid of the restrictions and adjust the JOIN in the base query. As @a_horse expressed it so poetically in the comments above:


  the perils of ORM frameworks


Anyway, to solve the question at hand:
After selecting rows where BASE.id and BASE.start_date / BASE.end_date match, we have four different cases:


No child at all, so CHILD.id is NULL.
Child matches.
Child does not match and no other matching child.
Child does not match but other matching child.


This is hard to solve on one query level. You could do it with window functions and CASE statements and DISTINCT, in fact I had a basic working query, but it gets monstrous.

Using a CTE dramatically simplifies the syntax.
In the CTE:


Filter BASE on id and date range.
Still includes rows with non-matching children.
Assign aliases for identical column names
Tag rows where child matches the date


In the final SELECT:


Return all rows with matching children
Return one row where no matching children exist, with NULLs for child-columns.


Here is working demo on sqlfiddle.com for those who want to toy with it.