Deduplicate SELECT statements in relational division
I see a possible minor simplification with a CTE (available in Redshift) for the repeated subquery with identical predicates:

WITH p2 AS (
   SELECT event_id
   FROM   params 
   WHERE  key = 'utm_source' AND value = 'facebook'
   )
SELECT e.visitor_id, '1'::text AS filter   
FROM          p2 
JOIN   params p1 USING (event_id)
JOIN   events e ON e.id = p2.event_id
WHERE  p1.key = 'utm_campaign' AND p1.value = 'campaign_one'
GROUP  BY e.visitor_id

UNION ALL
SELECT e.visitor_id, '2'::text AS filter   
FROM          p2 
JOIN   params p1 USING (event_id)
JOIN   events e ON e.id = p2.event_id
WHERE  p1.key = 'utm_campaign' AND p1.value = 'campaign_two'
GROUP  BY e.visitor_id;


Plain joins may also be faster than multiple IN semi-joins.

This multicolumn index should allow index-only scans on params:

CREATE INDEX foo_idx ON params (key, value, event_id)


Add another index on just (event_id) if you don't have one, yet.

Arsenal of available query techniques for relational division under this related question on SO:


How to filter SQL results in a has-many-through relation


Avoid multiple SELECTs

As commented by @Andriy, we can squeeze out some more:

WITH p2 AS (  -- repeated, immutable filter
   SELECT event_id
   FROM   params 
   WHERE  key = 'utm_source' AND value = 'facebook'
   )
, p3 (value, filter) AS (  -- values for variable filter
   SELECT text 'campaign_one', text '1'
   UNION ALL SELECT 'campaign_two', '2'
   )
SELECT e.visitor_id, p3.filter   
FROM          p3
JOIN   params p1 USING (value)
JOIN          p2 USING (event_id)
JOIN   events e ON e.id = p2.event_id
WHERE  p1.key = 'utm_campaign'  -- repeated for p1
GROUP  BY 1, 2;


In Postgres we could use a shorter, faster VALUES expression, but that feature is not currently supported in Redshift:

...
, p3 (value, filter) AS (
     VALUES
     (text 'campaign_one', text '1')
   , (     'campaign_two',      '2')
   )
...


For just two SELECT in the UNION, this won't buy much. But it should be a substantial improvement for more - like you mentioned.

2nd query without CTE

The CTE is not necessary for the second query. You can simplify to:

SELECT e.visitor_id, p3.filter   
FROM  (
   SELECT text 'campaign_one' AS value, text '1' AS filter
   UNION ALL SELECT 'campaign_two', '2'
   ) p3                            -- values for variable filter
JOIN   params p1 USING (value)
JOIN   params p2 USING (event_id)
JOIN   events e ON e.id = p2.event_id
WHERE  p1.key = 'utm_campaign'     -- repeated, immutable filters
AND    p2.key = 'utm_source'
AND    p2.value = 'facebook'
GROUP  BY 1, 2;


About CTEs

Resources for Common Table Expressions (as requested in the comment):


The manual.
Tag info for CTE.


Data-modifying CTEs are particularly useful. Example:


How do I insert a row which contains a foreign key?


Basics with an added advanced example on this related answer on SO:


Are SELECT type queries the only type that can be nested?