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Welcome to Just Answer and thanks for your question. If you are asking about the typical home shredder that is smaller than a waste basket, there is not that much dust created to be a health hazard just shredding bills, etc. Someone working in an industrial shredding facility would be smart to wear a dust mask, and there would be industrial dust-handling measures in place to make sure dust content of the air met OSHA standards.
Dust from a cross-cut paper shredder should be much less than is stirred up from dusting your furniture or sweeping the floor. It is also a device that is used intermittently, not constantly. Of far greater concern is the smoking which is known to be related to COPD and lung cancer. Your wife should see her doctor to determine why she is coughing so much. She can always wear a mask while using the shredder and see if that helps. There are no studies that have shown and increase in health problems from occasional use of a home shredder.