ONE of Augustus Pugin's many jobs in his teenage years was to scurry around the rafters at Covent Garden, pulling on ropes to produce the extravagant illusions that the theatre-goers below were so fond of. It was dangerous work, high up in the dark and with heavy flats moving at high speed. There were dangers after the show was over too. In early 19th-century London, prostitutes used the empty boxes for business and the air of licence that came with the theatre appears to have got to anyone who hung around it.

1) Can 'prostitute's here mean 'corrupt artists'?

2) Does 'used the empty boxes for business' mean 'they begged money'?

3) What is 'the air of licence'?

4) 'have got to anyone who hung around it' -> How can a noun, not a verb, be located next to 'have got to'?

Actually, I think there's nothing I can understand in this sentence.
1. No: it's plain whores from the city
2. No: they had sex with their customers (or just attracted them) in those theater boxes. Look in a dictionary for box.
3. Libertine/debauched atmosphere.
4. The sentence is right.

You just have to read more.
Let me rephrase the entire sentence for you.

Women who took money for sex (postitutes) used the empty boxes to have sex with their customers (their busines), and the knowledge of such scandalous behavior and the atmposphere of scandal (the air of license) appears to have affected (got to - applied to) everyone who spend time at Covent Garden.
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Ahnthe theatre appears to have got to anyone who hung around it.
Means to have affected or influenced.
 BarbaraPA's reply was promoted to an answer.
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I understand. Thanks you for all your help.