Is work done by how much velocity is changed or how much displacement is done? 
A sailboat is moving at a constant velocity. Is work being done by a net external force acting on the boat?

The answer key is "No" according to the work energy theorem about work is done when there is a change in final and initial velocity. But if I use Work = Force x displacement, even though the velocity is constant there is still some displacement done by the (constant) force acting on it. So I figured that the answer is "Yes". Which one is true?
 A: The wind is certainly doing work, because it applies a force and the point where the force is applied is displaced. However it isn't doing any work on the boat, it's doing the work on the water.
The key point is that the net force on the boat is zero. We know the net force on the boat is zero because the boat is moving at constant velocity - if the net force were non-zero the boat would be accelerating. Since the net force on the boat is zero no work is being done on the boat.
The velocity of the boat is constant because the drag of the water is balancing out the force applied by the wind. So overall the wind is applying a force to the water - the boat is just the instrument through which the force from the wind is communicated to the water. So the wind is doing work on the water, but not on the boat.
A: 
A sailboat is moving at a constant velocity. Is work being done by a net external force acting on the boat?

This is a bit of a trick question it tricks you into thinking there is a net external force. The boat is moving at a constant velocity; that's a given. That means that the net external force on the boat must be zero. 

But if I use Work = Force x displacement, even though the velocity is constant there is still some displacement done by the (constant) force acting on it.

Using work = force times displacement yields zero work because the net external force is zero.
You can decompose the work done on the boat into the work done by individual forces acting on the boat. Then you'll find that the boat's engine (or the paddler, or the wind) is doing positive work, but drag is doing negative work. They sum together to form zero. The net work done on the boat is still zero, no matter how you attack the problem.
A: Good one. 
The net forward thrust afforded by power of the engine, wind, rowing etc., is constantly overcoming and exceeding the retarding force of resistance by water, air, etc. 
So, the movement of boat in water and air is constantly impeded or constrained by water and air resistance-tending to bring the boat to halt. 
The forward thrust is constantly applied at the right measure to keep the boat moving ahead at constant speed.
So, work is done to change the velocity (FROM tending to stop TO constant forward motion). It implies that displacement happens.
A: The first thing I should point out is W$\cdot$x. x is the displacement. Direction is important.
That's really not the most important thing for your answer though. The punch line, Work is done if there's a force (hence change in velocity) and it causes displacement So, if something's moving a constant velocity, it doesn't have any force acting on it (effectively). So the displacement doesn't matter.
To answer in words of the question that was, Change in velocity does the work. Change in velocity always leads to displacement.
