I couldn't find it in any dictionary. Google News didn't help me. 'Make air' (by analogy with 'make an impression') would sound comical. 'Have' doesn't have the meaning I seek to convey. So what verb to use with 'respectable/pompous etc air'?
A casual phrasing:
He gives off an air of confidence
He exudes an air of confidence
Something I think is worth mentioning is that I'm not sure the whole "air of" phrasing works well with "respect". To say someone gives off/exudes an "air of respect" or a "respectable air" sounds more like a weird way of saying that someone is very respectful. But I'm guessing you mean that the person has a sort of aura about them that makes everyone else respect them. In that case you can say:
He commands respect
To "command" respect implies that you comport yourself in such a way that others feel compelled to treat you with respect, as opposed to just insisting that everyone respect you automatically ("demand respect").