Just like gear ratios, if you change the diameter you change the torque. The power will be the same, but if you decrease the primary gear the RPM will go up and the torque will increase.
I always remember it from riding a bicycle.When you change from the larger sprocket on the pedal gear to the smaller it is easier for you to pedal but you go slower.
In industrial application the shaft diameter is not often changed except to compensate for strength or weight. What is changed is the output shaft gear or pulley. To increase torque you decrease the drive pulley / gear or increase the loaded pulley / gear.
This is often done with constant speed motors to change the speed and torque of the output. If you look at the top of a drill press you will often see a bunch of stacked pulleys connected by a drive belt. Scroll down about half way on the page I linked below to see images of a drill press pulley system. They use 3 sets of pulleys and 2 belts to give more control over the speed, it works kinda like changing gears in a car.