As a crest moves through a point, in what ways is the motion of that point like the motion of a ball thrown straight upward?
A crest in a transverse wave causes displacement perpendicular from the direction of energy transfer. The particles in a transverse wave (and here, I assume a matter wave like water or earth because light is a different subject altogether) move up and down as the energy is transferred to the right or left.
Since a particle in the medium is oscillating perpendicular to the energy transfer, it moves up, away from the equilibrium line and down to the equilibrium line and down below the equilibrium line and back up again.
This is exactly like throwing a ball upward. It moves upward with some velocity and some kinetic energy and as it rises, it loses kinetic energy and velocity until the velocity is at zero, and then it heads downward again. For material waves, gravity is the factor that causes this.
There must be two conditions, you can use any of them. The first is when the crest reaches its maximum height, its velocity becomes zero, but acceleration remains zero. Although it is also same for the ball when ball reaches its maximum height, its velocity becomes zero but acceleration remains the same...
In the second case, you can use kinetic energy and potential gravitational energy. When they are at zero level, there is no potential gravitational energy, but there must be a kinetic energy. When they are at highest level, there must be a potential gravitational energy but not a kinetic energy. So in every state mechanical energy remains the same, because at bottom enery is transferred to kinectic energy, and at the top from kinetic energy it's transformed to potentinal energy.