Some will, some won't.
Most self respecting engines have electronic motor management these days, to regulate output voltage and frequency. Often you see really decent machines built into a sea container, extremely silent and the only way to tell that a generator is in them is because you're all of a sudden walking through a beam of hot air. These containers work as a Faraday cage, which in turn will probably make them insusceptible again.
The cheap and tiny ones are often consumer stuff, they can't afford electronic management and are unstable in both voltage and frequency aswell as depending on load characteristic.
On the other hand ... it all depends on the energy contained in the EMP ... I guess you can never be really sure. Don't know if EMP is viable to melt generator windings ...
The simplest way to do this would be to simply put LEDs in series with the generator. There are DC generators that wouldn't require a rectifier, and in fact, any DC motor is also a DC generator. As the generator turns faster, the current through the LEDs increases, and the generator becomes harder to turn. Depending on the rating of the LEDs, the efficiency of the generator, and how hard you turn it, you may overpower the LEDs. Putting a resistor in series with them will dissipate some of the power as heat.
If you wanted the LEDs to continue glowing after the generator is no longer turning, you could add to this circuit a capacitor in parallel. You will want a resistor in series with the LEDs so that the LEDs don't clamp the voltage across the capacitor. As you turn the generator the LEDs will glow. Also, current will flow through the capacitor to keep its voltage equal to the voltage generated by the generator, and in doing so, energy will be stored. When you stop turning the generator, this voltage will persist, which will light the LEDs and also keep the generator spinning, until the capacitor has no energy left. I haven't done the math, but intuition says you will need the biggest capacitor you can find.
To prevent the capacitor from turning the generator, thus reserving more of its energy for the LEDs, you can put a diode in series with the motor. You could even make it an LED, which is a kind of diode, to visualize the flow of electricity. Here's the circuit so far:
It's hardly efficient, but it is simple.
Best Answer
It doesn't have to be a switcher. With 120V AC you could use an AC transformer to get 12V AC at (somewhat less than) 58 amps, then a rectifier, reservoir capacitor and (if necessary for your application) voltage regulator to get 12V DC. (I'm simplifying a bit: unregulated, 12V AC would give closer to 15V DC)
A PC power supply will be easier to find at 20A+ ratings, but you may find e.g. a car battery charger that would do the job.
My main worry would be that the AC from a small generator might be too poorly conditioned/regulated for a PC power supply and cause it to fail. I might be overcautious here, but the transformer/rectifier solution is likely to be more robust. You don't say what the load is; knowing that may help improve the answers.