Is it safe to work in a cooler/freezer when you have asthma?

Depends... Some people will have asthma exacerbation when breathing cold air. If this happens, then this is not the right job for that person. If the person tolerates cold air, then it is ok.
Maybe not. Cold air is a significant trigger for some people with asthma. Using albuterol before starting work each day or taking a controller medication like an inhaled corticosteroid of singulair (montelukast) may protect your lungs. Still this may not be the best career choice for you.