Safe connection of a 4pin Fan to an Arduino
If the fan behaves according to the norm, you can drive its PWM pin with an Arduino. 

The PWM pin accepts 5V input, and sources no more than 5mA, so it should be fine. Many fans should work with Arduino´s default PWM. The Norm says you have to use 25 kHz PWM frequency though, which is a bit tricky. See AKA's post in the Control PWM Fan Thread.

Yes, Arduino and the 12V power supply for the fan need to have common ground. 

PWM 4pin Norm

Control PWM Fan Thread

moderator note:
This answer have arrived to this thread as a result of a merge. The 4-wire fain in the original thread: AK-FN073.