There are several aspects that you have to consider. If the size of the balancing robot is small, you can easily control two 5V small dc motors directly with arduino/RPi-3. But, if the motors are big, then you would need 2 big motors (12V) and you would need a Raspberry Pi 3 + motor controllers or Arduino + motor controllers.
Your choice of control algorithm (LQR, PID, etc) will affect the type of microcontroller board that you shuold use. But, Raspberry Pi-3 should be able to do the job as Arduino is lacking in the computational power when running complex algorithm.