The only bill that I can think of that was passed during Woodrow Wilson's second term and that would have showed that his government was trying to legislate morality was the 18th Amendment. This was ratified in 1919. The Volstead Act, passed to enforce Prohibition, could also be categorized in this same way.
This amendment, of course, is the one that brought about Prohibition. Prohibition was perhaps the most prominent example in all of American history of an attempt to legislate morality. The idea behind Prohibition was, in large part, that drinking alcohol was bad for people and that they should be protected from their destructive desire to drink. By basing such a sweeping law on this sort of idea, Wilson and Congress were clearly trying to legislate morality.