Willy Taveras has now started 6 World Series games ... and is 0-6. I took a quick look, and for active players, I believe that's the most.
Going back through history, sweeps are pretty rare, only 17 of 102 from 1903-2006. The Cubs were swept in 1932 and 1938, but it all of those hitters that played in both '32 and '38 also played in the 1935 Series where the Cubs won 2 games.
There is a good chance Willy T. has currently started the most World Series games in history without a single victory. 5 of those 6 games have been decided by 2 runs or less.
Russ Springer has been on the roster for 8 World Series games and is 0-8, but he only appeared in 4 of them and of course, started none of them.
