Which kind of motor is best recommended for a curtain opener?
A DC motor is likely up to the task, having functionality for up, down, stop, and naturally automatic stop at top and bottom with some limit switches or current sensing.  You can get DC motors of all sizes and use a gearbox in addition to whatever pulleys the system requires if necessary to provide more torque and voltage/pwm control to control speed.

If you are making "smart curtains" and require precise digital control of the curtain's position, that would be the application for a servo or stepper motor.  Something similar could be achieved by calibrating control of a DC motor, but it wouldn't be precise.  Servos and steppers require more complicated control mechanisms, but if the device had a microcontroller in it anyway, that might not be a big deal.

First thing you should figure out is roughly how much torque you need to move the curtains at the greatest speed you will want to.  You can use that to start comparing the abilities of motors in that class.  The voltage you have readily available may be a factor too.