Is there a word for "air can pass through it"?

permeable if the pores are small

allowing liquids or gases to pass through; capable of being permeated; penetrable; especially: having pores or openings that permit liquids or gases to pass through

breathable if it's a fabric

Permitting air to pass through: a breathable fabric.

I don't know what to call this property in a screen door...

apologies to @James McLeod - I didn't read the entire question and jumped on permeable.

Rather to my disgust (as some who grew up reading a great deal of science fiction), the industry-accepted term for this is "breathable."

I know this is ugly and misleading, but it's the word.

In addition to other answers' suggestions of breathable and permeable (admittedly better suited to your use case), I'd add porous for some uses:

(of a rock or other material) having minute interstices through which liquid or air may pass.

For example, to describe a fabric as one that air can pass through, you can use "breathability".

Quoting from wikipedia: Breathability is the ability of a fabric to allow moisture vapor to be transmitted through the material.

Air Permeability is the ability of a fabric to allow air to pass through it. While Air Permeable fabrics tend to have relatively high moisture vapor transmission, it is not necessary to be Air Permeable to be breathable.

Hope this helps one way or another.