The first sentence, Be at the airport... is an imperative sentence; you tell someone that they should do something. However, it can also be interpreted as a non-finite clause (see below). The correct analysis depends on the context. Here is an example as an imperative:
"Be at the airport to meet him!"
"OK, I will be there."
The second sentence is a non-finite clause (Being at the airport...) followed by an infinitive clause (to meet him...). It does not contain a finite verb, and so is not a fully grammatical sentence. It could be used in conversation as a reply to a question, as in spoken language you often infer parts of the sentence from context. But even then the second sentence still sounds a bit odd, and you would typically prefer the first sentence:
Q: "What did I ask you to do?"
A: "Be at the airport to meet him."
Q:"What was your task for today?"
A: "Being at the airport by 4 o'clock to meet him."
You would infer "You asked me to..." in the first sentence, and "My task was..." in the second one.