Is it safe to use a variable AC transformer to control the speed of a box fan?
Most portable fans use a type of motor called a "shaded pole AC induction motor".  Here is a description of them from Wikipedia:

Shaded Pole Motor

You should not, however, try to control the speed using a "variac" device because the motor will attempt to continue to try to draw the same amount of POWER (Power = Voltage * Amperage) and you will get heating and lead to premature failure or even a fire hazard.

Here is a link to a discussion of low-voltage issues with AC motors:

AC Motor Low Voltage Issues

To quote the relevant section:


  Effects of low voltage. When you subject a motor to voltages below the
  nameplate rating, some of the motor's characteristics will change
  slightly and others will change dramatically. To drive a fixed
  mechanical load connected to the shaft, a motor must draw a fixed
  amount of power from the line. The amount of power the motor draws has
  a rough correlation to the voltage 2current (amps). Thus, when voltage
  gets low, the current must increase to provide the same amount of
  power. An increase in current is a danger to the motor only if that
  current exceeds the motor's nameplate current rating. When amps go
  above the nameplate rating, heat begins to build up in the motor.
  Without a timely correction, this heat will damage the motor. The more
  heat and the longer the exposure to it, the more damage to the motor.


Fans with a built-in speed control generally have multiple windings on the motor that are switched in and out in order to control the motor speed.  These will not suffer from the overheating effect.

Your best bet here would be to get a different fan that can be controlled to your liking.