I think that the usual rational is that allowing it enables people being rewarded or punished based on their votes. In particular, prohibiting it prevents people being pressured by spouses or employers to prove they voted the "right" way.

What Evan said. There was a lot of pressure to vote "the correct way". Lincoln got no votes in the South because everyone could have seen you trying to vote for him. The secret ballot wasn't adopted in the US until the 1880s-90s. Partisan premarked ballots were used until 1950! ->