In 1911 Larry McLean (Fredericton, NB), a catcher for CIN-N threw out 14 consecutive baserunners trying to steal! This was accomplished during a 5 game span from 9-2-1911 to 9-6-1911. (Info from Retrosheet). During those 5 games he also picked off 2 runners in addition to the 14 CS. Also, in a span of 9 games that he played from 8-28-1911 to 9-7-1911 he had a total of 2 SB allowed with 20 CS & 2 PKO!
I would imagine that the 14 CS in a row is probably a record, although of course data is very spotty in this category.