How generator designers determine the power factor?

The generator designers will have to determine the winding cross section area and specific current/mm² to satisfy the required current, and they will have to determine the required total flux and flux variation per unit of time per winding to satisfy the voltage requirement. Then they will have to determine how the primary flux source will be generated (excitation), and how any required mechanical power can be transmitted into the electro-mechanical system, with the appropriate speed for the required frequency.
In all the above, we can have parallel paths of current, as well as of flux, in all sorts of combinations.

1) All ordinary AC power depends on electrical induction, which basically is flux variations through coils of wire. (In the stator windings).
2) Generator rotor current (also called excitation) is not directly related to Power Factor, but to the no-load voltage generated.
3) The reason for operating near unity Power Factor is rather that it gives the most power per ton of materials used in the generating system, and at the same time minimises the transmission losses.
4) Most Generating companies do charge larger users for MVAr, and for the private user, it is included in the tariff, based on some assumed average PF less than unity.
5) In some situations, synchronous generators has been used simply as VAr compensators, with zero power factor. They are much simpler to control than static VAr compensators, can be varied continuously, and do not generate harmonics. Unfortunately they have higher maintenance cost.
6) When the torque from the prime mover exceeds a certain limit, it can cause pole slip. The limit when that happens depends on the available flux (from excitation current), and stator current (from/to the connected load).