Someone Standing At The Edge Of A Cliff Throws A Ball Nearly Straight Up At A Certain Speed And Another Ball Nearly Straight Down With The Same Initial Speed. If Air Resistance Is Negligible, Which Ball Will Have The Greater Speed When It Strikes The?

Both the balls will strike the ground at same speed.

This answer  was not obvious immediately.
When you stand on the cliff both balls have some potential energy.with that potential energy you are adding some kinetic energy.

Consider the first ball which was thrown in upward direction.with the initial potential energy of the ball you are adding some kinetic energy.when the ball move upwards,the kinetic energy of the ball was converted into potential energy.when all the kinetic energy was converted into potential energy,the ball stops moving upwards and accelerate downward.Now the potential energy of the ball was gradually converted into kinetic energy.when the ball reaches the height equal to the height of the cliff,the ball will have kinetic energy equal to the initial kinetic energy given by you to the ball but in downward direction.This is same as the second ball which has same kinetic energy and moves downward.since the kinetic energy was equal,the velocity of the two balls was also equal when they reach the ground.