I am guessing I can run it right off the back of the alternator with the protection of a fuse?
No, because if your generator has a 12v alternator on it, that alternator really should be feeding a battery, such as a separate battery for starting the unit. So connecting the fan there will have it running full time, and running the battery down.
I'm with Clifford on this -- you really want a 120-VAC fan. In which case, you simply connect it to the generator main output by way of an individual breaker appropriately sized for the fan. However, if you really want to try to run with the 12v fan, you have two options:
1. Connect it directly to the battery/alternator setup by means of a 120-VAC relay. Connect the relay coil to the main generator output through a small breaker.
2. Connect it to the 12v "run" circuit of the generator wiring, assuming that circuit is large enough to handle the load. You'll need the wiring diagram for the genny to find this; it usually runs through the safety switches (temp and oil pressure) and then on to the fuel solenoid.