Sizing a generator to run a welder is tricky because welders have huge inrush current. The welder may be rated to draw 30 amps (for example) at max output, but for a fraction of a second when you strike an arc, it may draw 100 amps (for example). On household current, this is no problem, but a generator will bog down. When that happens, the voltage drops and you lose your arc. The exact specifics depend on too many factors to account for in a forum post, but the bottom line is that you should test out your welder on a generator before buying. Be sure to measure voltage on the line while welding, to be sure that you're not running in an under-voltage situation. In many cases, it becomes more cost-effective to buy a motor-driven welder and use it as a generator, than to buy a big enough generator to run your mains-powered welder.
BTW, I have a personal theory, that I have yet to test out, that if you are doing pedal-controlled TIG, or 4T TIG, that this problem goes away, because you can strike the arc at a very low amperage and then slowly ramp up in a controlled manner that won't exceed the generator's abilities. But stick and MIG are another story.