Theodore Roosevelt's "stewardship" theory was based off the idea that the president could do anything that the Constitution did not forbid. He believed the president to be a "steward" of the people and "to do all he could for the people." By exercising this type of power Roosevelt believed he was able to do what was necessary for the country and what it demanded. To watch out for the "common well-being" of the nation, Roosevelt believed this type of informal power had to be utilized.
William Taft's "literalist" theory was based upon the idea that the president could not do anything that the Constitution did not permit. He believed that a president's "jurisdiction must be justified and vindicated by affirmative constitutional" provisions, or else those powers do not exist. Taft argues that by giving the executive informal powers is not only limitless, but also unsafe, creating the potential to invade upon private right.
Both Taft, and Roosevelt, make valid points in their views on presidential power. I think that many people would flip flop between the two theories depending upon the president that held office, but in a general sense, I would tend to side with Roosevelt's argument. Certain circumstances call for certain actions which could not have been fathomed by the Framers, so by limiting executive power to that of which only resides in the Constitution would be an unwise call. Times of emergency, like war or natural disaster, call for hasty actions that may not be constitutional provisions. By giving the president room to move outside of the Constitution provides for a better working democracy. Everything the people demand is not going to be within the limitations of the Constitution.
Going back to Taft's argument, I believe that a president's informal powers must be limited as well. Since there is no clear definition of what the common good is, how can anyone really know if a president's actions are in accordance with this ideal? For example, presidents like Richard Nixon, who obviously overstepped their boundaries and abused executive power, need to have some limitations, but it's hard to draw the line where. If a president's power is too limited people may be unhappy with the government's abilities, but if it is loose it runs the risk of being too powerful. I think overall it is a tough question as to where to draw the line.
Ultimately, as I said before, Roosevelt's argument probably makes the most sense, of course with certain limitations. Just because something falls somewhere in between what is legal and what is illegal doesn't always make it right. There can be deviance without having broken laws.