How do generator designers determine the power factor for their generator(s)? Do they have to carry out experiments or could they simply compute directly based on the electrical components being used?

Power factor is not determined by the generator, but by the load!

Oops, you are right. Generators do not have a power factor but are rated in KVA. I was thinking about hydroelectric power generation when I wrote the question.

Power factor changes with load and also excitation. Excitation controls the Voltage. Hence Generator should not be run at what ever power factor is demanded by the load. Poor power factor can cause over heating and power factor close to unity can cause machine instability. Hence the generator power factor has to be adjusted to maintain the machine in stable region on the capability curve of the machine.

Sorry, but generators can in general safely be operated at any power factor, even negative. What limits the operation of a generator is the stator current, and the generated, or applied voltage.
When we talk of instability, it is either caused by too low voltage, or excitation, or too high torque supplied from the main driver.
We can also have instability in cases where the AVR or the governor is "hunting" due to incorrect settings corresponding to the characteristics of the connected network (load).

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.