Which airliners have carried the most passengers?
As you note the 737 and A320 have a huge leg up on most other aircraft simply by production numbers. The 737 is the unquestionably the most produced large passenger jet aircraft in history when compared to the A320 it has 20 years of extra flights under its belt which is a lot of time to move a lot of people. Its a bit dated but Boeing provides a bit of data on this and estimates:


  • The 737 family has carried more than 16.8 billion passengers; that
  is equivalent to every single man, woman and child flying at least
  twice. (2013 world population was
  7.1 billion).*
  
  *As of December, 31, 2013


Airbus provides a similar stat estimating: 


  15,6+ billion passengers carried since Entry Into Service
  
  As of November 2018


So it seems that the 737 is clearly in the lead over the A320 as 5 years ago they had already carried an estimated Billion more passengers than the A320 has as of last month. 

These are the only two large commercial jet aircraft that even appear on the most produced aircraft list linked above. Other sources quote the 747 as carrying 3.5 Billion passengers, the 727 comes in around 2.2 Billion in its lifetime and the 757 is somewhere in the 1.3 Billion passenger range which is similar to the A330's 1.3 Billion (as of 2013) none of which even approach double digits so the 737 is clearly the leader from the estimates I can find.