I echo what others have said.
While budgies (small parakeets) are especially sensitive to drafts, the real culprit for all tropical birds is more the 'temperature' of the wind in relation to the temperature of the room they are in. A cold winter draft from a window/door into a 70degree room is usually the biggest culprit in causing illness in tropical birds from drafts.
I also think, at least in my experience, that a small draft that only lasts for a couple seconds won't do immediate harm for healthy birds. But if left exposed to the draft for a prolonged period, then you get trouble. But precaution should be taken to avoid any draft of significant temperature change. (ie, a 25degree draft in a 70degree room).
"Room temp" wind, like from a standard fan is not a danger at all, provided that the ambient room temp is appropriate. In fact, many parrots like those.
My cockatiel Peaches, and my sun conure Skittles, both like to sit in front of the fan and will often close their eyes and just sit there.
I tend to pay special attention to winter drafts (living in Maine, I have too). I also pay close attention to the room temp when Skittles is wet from a bath.
You should just pay attention to your birds body language.
But the health of your bird has as much an impact on illness as temperature fluctuations. An unhealthy bird (not adequately nourished, ie Vit.A def.) is much more susceptible to catching illness than a healthy fed bird. Just as is the case with humans. Your body is much more capable of fighting off ailments when you are already in good health, nutrition wise.