Answer
No, the escape speed doesn’t depend on the direction of launch.
Yes, the answer depends on whether one includes the effects of air resistance.
Work Step by Step
No matter the direction of launch, the end result is the same, with the object being very far from the Earth and having zero gravitational potential energy. The derivation of escape speed (13.29) ignores the initial launch direction. The escape speed doesn’t depend on the direction of launch.
If one includes the effects of air resistance, mechanical energy decreases because of the negative work done by air resistance, so the object must be given a larger initial kinetic energy to compensate for the drain. In other words, the escape speed increases.
