Use DEFAULT value if empty string

A check constraint will only prevent putting such a value into the column. It will not magically replace the supplied value with the default value.

In order to silently replace an empty string (or null) with the value 'active' you will need a trigger:

create or replace function check_default()
 returns trigger
as
$body$
begin
   if (new.status is null or new.status = '') then
      new.status = 'active';
   end if;
   return new;
end;
$body$
language plpgsql;

The above trigger would catch stuff like this:

insert into foo (id, active) 
values (42,'');

update foo 
  set active = ''
where id = 42;

You could even extend that logic to also replace whitespace only values (' ') with the desired value.

Although it is possible to retrieve the default value dynamically in the trigger (to avoid having the same constant in two places) I would not do that for performance reasons.