In the healthcare industry, researchers have been developing machine learning models to automate diagnosing patients with respiratory illnesses based on their breathing patterns. However, these models do not consider the demographic biases, particularly sex bias, that often occur when models are trained with a skewed patient dataset. Hence, it is essential in such an important industry to reduce this bias so that models can make fair diagnoses. In this work, we examine the bias in models used to detect breathing patterns of two major respiratory diseases, i.e., chronic obstructive pulmonary disease (COPD) and COVID-19. Using decision tree models trained with audio recordings of breathing patterns obtained from two open-source datasets consisting of 29 COPD and 680 COVID-19-positive patients, we analyze the effect of sex bias on the models. With a threshold optimizer and two constraints (demographic parity and equalized odds) to mitigate the bias, we witness 81.43% (demographic parity difference) and 71.81% (equalized odds difference) improvements. These findings are statistically significant.