I like him just fine historically, but I don't think he can hold court with the other faces on that monument, just my opinion. But if I were to make a case for him, I can argue that only he and Jimmy Carter were Presidents that made serious efforts towards establishing and maintaining peace. His Fourteen Points, while a failure diplomatically for the most part, were genius and revolutionary for the time.
An idea of Freedom of the Seas, for example, was foreign to most imperial wisdom of the time, and today it is commonly accepted international law. His League of Nations, which seemed pie in the sky optimistic (and it probably was) at the time, was the forerunner of the United Nations today which, while it still has issues, has done a lot of good in this world over the past 65 years.
So Wilson avoided war as long as possible, then tried to cure the world of war once it was over, and I'd have to give him some credit for that, even if not a face on Rushmore.
I'm not sure that I would attempt to squeeze Woodrow Wilson's image onto the already crowded Mount Rushmore, but Wilson most certainly should be considered one of the greatest presidents in United States history. An intellectual and former president of Princeton University, Wilson took advantage of the divided Republican party (whose leaders William Howard Taft and Theodore Roosevelt were both on the ballot) to become the 28th president. Among his accomplishments were the passing of the Federal Reserve Act; the organization of the Federal Trade Commission; and Revenue Act of 1913 which gave Americans their first progressive income tax. He guided the U.S. through World War I, negotiated the armistice and Treaty of Versailles, and helped to create the League of Nations. For these accomplishments, he was awarded the Nobel Peace Prize. His policy of "Wilsonianism," the active spreading of democracy internationally, has dictated American foreign policy ever since. It can also be argued that Wilson sacrificed his own health during his duties as President, suffering a terrible stroke in 1919 that robbed him of the ability to walk unaided.
I do not know that all that many people would say that President Wilson should be on Mt. Rushmore. But if they did, I suppose the argument would go something like this:
Wilson was a very important president because he was president during World War I. He tried hard to win the war. But most importantly, he tried to remake the world after the war. Wilson's fourteen points, and his attempts to create the League of Nations, were major efforts to achieve peace.
In addition, Wilson was one of the Progressive presidents who helped bring about new laws that protected regular people from big businesses.