The children remembered Mr. Poe because he always had a cold and was
constantly excusing himself fromthe table to have a fit of coughing in the next room.

The top quote is from Book 1 (The Bad Beginning) and the next is from Book 2 (The Reptile Room).

The driver of the car was Mr. Poe, a family
friend who worked at a bank and always had
a cough.

It seems he is just sickly, and is constantly at a state of having a cold–which seems like a rather unfortunate event to me. Or he has constant year-round mild allergies and hasn’t found a fix for them yet.