How do I use properly CASE..WHEN in MySQL

Remove the course_enrollment_settings.base_price immediately after CASE:

SELECT
   CASE
    WHEN course_enrollment_settings.base_price = 0      THEN 1
    ...
    END

CASE has two different forms, as detailed in the manual. Here, you want the second form since you're using search conditions.

CASE case_value
    WHEN when_value THEN statements
    [WHEN when_value THEN statements]
    ELSE statements
END 

Or:

CASE
WHEN <search_condition> THEN statements
[WHEN <search_condition> THEN statements] 
ELSE statements
END 

here CASE is an expression in 2nd scenario search_condition will evaluate and if no search_condition is equal then execute else

SELECT
   CASE course_enrollment_settings.base_price
    WHEN course_enrollment_settings.base_price = 0      THEN 1

should be

SELECT
   CASE 
    WHEN course_enrollment_settings.base_price = 0      THEN 1

CASE course_enrollment_settings.base_price is wrong here, it should be just CASE

SELECT 
CASE 
WHEN course_enrollment_settings.base_price = 0      THEN 1 
WHEN course_enrollment_settings.base_price<101      THEN 2 
WHEN course_enrollment_settings.base_price>100 AND    
                  course_enrollment_settings.base_price<201 THEN 3 
    ELSE 6 
END AS 'calc_base_price', 
course_enrollment_settings.base_price 
FROM 
  course_enrollment_settings 
WHERE course_enrollment_settings.base_price = 0 

Some explanations. Your original query will be executed as :

SELECT 
CASE 0
WHEN 0=0 THEN 1 -- condition evaluates to 1, then 0 (from CASE 0)compares to 1 - false
WHEN 0<1 THEN 2 -- condition evaluates to 1,then 0 (from CASE 0)compares to 1 - false
WHEN 0>100 and 0<201 THEN 3 -- evaluates to 0 ,then 0 (from CASE 0)compares to 0 - true
ELSE 6, ...

it's why you always get 3