Can an Alar Reduction Compromise Breathing?

The short answer is very unlikely. Your nostrils are not the narrow part of your respiratory tract. That part is about an inch back from the nostril opening and is called 'the nasal valve'. Therefore the base of your nose can be narrowed, usually as much as needed without compromising your breathing. If the narrowing is excessive, then you would expect not only trouble breathing but an abnormal appearance.

Alar reduction typically refers to excisions of the nasal sill or nostril excisions to reduce nasal tip width.  This usually does not result in breathing difficulties since the external nasal valve is not compromised by this procedure.

Anything is possible but a correctly performed alar base reduction will doubtful give you any airway difficulty.

Alar base reduction will have no effect on breathing when properly done. Therefore, see an experienced rhinoplasty surgeon.

When performed judiciously, alar base modification should have no effect on breathing.  Typically the sensation of nasal obstruction or flow is most easily affected at an area referred to as the "nasal valve", which should not be affected by alar base modification.  Another way to explain this is that since the inlet of the nostrils - where the alar base modification occurs - is not the bottleneck to airflow, judicious narrowing of it will not compromise breathing.