How does the temperature of water affect the reactivity of alkali metals acting on it?
Increasing temperature does indeed increase the reactivity of alkali metals, or most things for that matter. The rate of reaction increases because the average kinetic energy of the molecules increases, and so more collisions can overcome the activation energy for the reaction to take place.

The reason for using cold water rather than hot water for the reactions of alkali metals is that alkali metals are already reactive enough at room temperature. Even fairly small lumps of potassium will burn if you put them in cold water so imagine what it would be like with hot water. This is not something you want to try without very careful precautions.