In addition, the Southern bank of the Saint Lawrence in Quebec today is far, far more populated than the northern bank. Look on a topographic map, and you'll see on the northern side the valley ends much more quickly - less good farmland.
Honestly, whenever I see a map with the Saint Lawrence as the border of the U.S., it makes me think the mapmaker is lazy and doesn't know anything about Canada. Rivers often form the borders of U.S. states because borders between states don't need to be defensible, and because they can form one contiguous cultural sphere despite the state border.
However, rivers are seldom international boundaries, especially in long-settled areas. Look at Europe. Only small bits of the Rhine, Danube, and Oder are international borders, and in essentially all cases these borders only formed within the last 150 years or so, have little to do with cultural geography, and often involved either forced assimilation or population exchanges.
As others have said, if Britain cedes Quebec south of the Saint Lawrence, they might as well give it all up. And neither side of the border will be very secure, since either the U.S. or the U.K. could send warships up the river and fire cannon with impunity. I could see the U.S. gaining the Eastern Townships, but that's about it.