PL/pgSQL checking if a row exists

Simpler, shorter, faster: EXISTS.

IF EXISTS (SELECT 1 FROM people p WHERE p.person_id = my_person_id) THEN
  -- do something
END IF;

The query planner can stop at the first row found - as opposed to count(), which will scan all (matching) rows regardless. Makes a difference with big tables. The difference is small for a condition on a unique column: only one row qualifies and there is an index to look it up quickly.

^{Improved with input from @a_horse_with_no_name in the comments.}

You can just use an empty SELECT list:

IF EXISTS (SELECT FROM people p WHERE p.person_id = my_person_id) THEN ...

The SELECT list has no influence on the result of EXISTS. Only the existence of at least one qualifying row matters.

Use count(*)

declare 
   cnt integer;
begin
  SELECT count(*) INTO cnt
  FROM people
  WHERE person_id = my_person_id;

IF cnt > 0 THEN
  -- Do something
END IF;

Edit (for the downvoter who didn't read the statement and others who might be doing something similar)

The solution is only effective because there is a where clause on a column (and the name of the column suggests that its the primary key - so the where clause is highly effective)

Because of that where clause there is no need to use a LIMIT or something else to test the presence of a row that is identified by its primary key. It is an effective way to test this.