How do I determine the proper input voltage of a high voltage transformer?

You need to either know your turns ratio or maximum current rating for each coil.
In any case, connecting 110v to a 120v generator or vice verse won't really cause too much damage.

The easiest way would be to put a lower voltage on your primary coil and measure the voltage on the secondary coil to get the turns ratio, but for a 3.1kV transformer that may require some heavy duty measuring equipment

One way to test the turns ratio is to apply a small voltage, such as 120 volts, to the high side.  Measure the induced voltage on the low voltage side, which should be about 4.5 volts.  You could also use a 120 : 24 Vac control transformer, the type used for hydronic heating zone valve control.  With such a control transformer, you would apply 24 volts to the 3 kV leads and all of the voltages and total energy would be well within safe limits in case the transformer was internally shorted.

Before applying rated voltage, it would be best if you could give it a megger test at 500V or 1000V to test the insulation.  Measure from each lead to the frame to test for leakage.  Minimum ohm reading should be about 5 kohm per rated volt.  So measurement on the input would nominally be several megohms but could be as low as 1 Mohm.  High side measurements should be a minimum of 30 Mohm.

When people say 110 V, they are rounding down 117 VRMS 60 Hz. AC to 110 V and, when people say 120 V, they are rounding up 117 VRMS 60 Hz. AC to 120 V. Therefore, it is all the same voltage and you cannot do damage to a transformer that is designed to be connected to the US power grid by connecting it to a wall outlet. However, be very careful to insulate the 3.1 kV secondary voltage from personal contact.

(VRMS = Volts Root Mean Square; it is a way of averaging the effective voltage of a sine-wave)

Transformers.  
A device for stepping voltages and impedance up or down. It depends on the ratio of the turns
of wire on the on the primary and the secondary coils.                            Np    Ep    Is             Zp
The formula for this is Np/Ns = Ep/Es = Is/Ip = sqrt. Zp/Zs.                  ---- = --- = --- =  sqrt.---
N, number of turns. E, voltage. I, current. Z, impedance.                         Ns    Es    Ip             Zs

Unless the transformer is old, 120 volt is more likely than 110 volts. Typically an 11% error in the input voltage only derates a transformer about 5%, so it is likely unimportant unless the transformer has a barely adequate KVA rating for your application. At light load the transformer likely produces 3.1 KV with 110 volts input so measuring the output voltage is a clue = expect about 3.5 KV at light load with about 120 volts input. Unfortunately your ac volt meter may have an error of 10% at about 3.2 KVA especially if the power company is supplying a slightly non-sinusoidal wave form. Please be careful your first shock at 3.1 KV may be the last thing that you ever do.  Neil