how can I motorize hand cranked standing desk
Let's say it takes 20 turns to complete the transition, and you want it done in 10 seconds. Then required crank speed is 120 RPM. 

Now, if you use 1:60 gearbox then the motor you are looking for is 7200 RPM with only 500 g-cm torque. There are many DC motors in this category, although this is rather on high side. If slightly longer time is OK with you then finding motor will be easier.

So, the design parameters for geared motor are: 80-150 RPM, 2.5 N-m (25 kg-cm)
For non-geared: 7000-15000 RPM, 0.04 N-m (0.42 kg-cm) + 1:60-1:150 gearbox

With all of the above start browsing. Some examples of what you can find:
24V, 30 kg-cm, 60 RPM, 1.6A
24V, 27 kg-cm, 90 RPM, 3.5A
12V, 30 kg-cm, 85 RPM, 5A - "window motor"

Automotive 12V power window motors seems to be great choice for this project.

Also note the following:


You don't need stepper motor. Because of high gear ratio it is easy to achieve precise positioning of the desk with simple DC motor.
It seems you don't need uni-directional (like worm gear) transmission either, because cranks like that usually have worm gear already, no risk of desk falling down when voltage removed. UPDATE: from your comment it is clear you need a worm gear after all.
You do not need Arduino or any kind of computer control. A simple sircuit with  either two DPDT buttons or one non-locking rocker switch is sufficient.
You do need two end switches to stop motor at the ends of travel range.


Here is an example of a circuit with two "up" and "down" buttons and two end switches. When I see somebody trying to use Arduino for something as simple as this I am losing my faith in humanity :)