The correct way to write this dialogue, would be: President Woodrow Wilson said that baseball is the national pastime.
Explanation:
The reason for this is that this is a form of reported speech, in which we are literally reporting the words that President Woodrow Wilson said. In direct speech, we would find this sentence in the following way: President Wilson: "Baseball is the national pastime." But because we are reporting what the character said, then the correct structure for this reported speech is to take the sentence in the question, which in essence is in the form of reported speech, and we just need to add the "that" between the "said" and the initiation of what the president actually said.
