When using a stair climber, some common mistakes to avoid include using improper form, not maintaining proper balance, holding onto the handrails for support too tightly, relying too much on the handrails for support, leaning forward or backward too much, going too fast or too slow, not engaging the core muscles, not wearing proper footwear, not adjusting the machine to the correct intensity level, and not properly warming up or cooling down before and after using the machine.