CASE statement multiple conditions
Just add a WHERE condition.  And if you are only displaying one value, RUNNING, then there is no reason for a CASE. Otherwise you will want to evaluate each condition in the CASE including what should display in the event none of the conditions is met; a default value.

So here is the code to your original question:

SELECT 'RUNNING' AS A
from message_log 
where  Message = 'BUILD' 
    AND Log = 'Day Start'
order by 1;


And here is the code for a multi-condition CASE:

SELECT CASE 
    WHEN (Log = 'Day Start') THEN 'RUNNING' 
    WHEN (Log = 'Day End')   THEN 'NOT RUNNING'
    ELSE 'UNKNOWN' END AS A
from message_log 
    where  Message = 'BUILD' 
order by 1;