The lady looked at him and at once dropped her eyes.
"He doesn't bite," she said, and blushed. (1.9-10)
Anna is immediately characterized by a sense of modesty; this renders her later indiscretion all the more shocking, and all the more indicative of a major shift in her character.
In the evening when the wind had dropped a little, they went out on the groyne to see the steamer come in. There were a great many people walking about the harbour; they had gathered to welcome some one, bringing bouquets. And two peculiarities of a well-dressed Yalta crowd were very conspicuous: the elderly ladies were dressed like young ones, and there were great numbers of generals. (2.2)
Chekhov doesn't let us forget the social context in which Gurov and Anna's affair takes place.
Then he looked at her intently, and all at once put his arm round her and kissed her on the lips, and breathed in the moisture and the fragrance of the flowers; and he immediately looked round him, anxiously wondering whether any one had seen them. (2.7)
Even in the lax atmosphere of Yalta, Gurov can't forget the restrictions of society and the threat his relationship with Anna poses to his reputation.
The hotel porter gave him the necessary information; Von Diderits lived in a house of his own in Old Gontcharny Street – it was not far from the hotel: he was rich and lived in good style, and had his own horses; every one in the town knew him. The porter pronounced the name "Dridirits." (3.15)
Gurov is definitely concerned with social status and reputation – through the narrative perspective we can see that he's judging those around him (in this case, the porter and his pronunciation).
"So much for the lady with the dog...so much for the adventure....You're in a nice fix...." (3.22)
Gurov recognizes that things have fundamentally changed since Yalta. If he is to lose Anna now, it must be under entirely different circumstances.
Anna Sergeyevna, too, came in. She sat down in the third row, and when Gurov looked at her his heart contracted, and he understood clearly that for him there was in the whole world no creature so near, so precious, and so important to him; she, this little woman, in no way remarkable, lost in a provincial crowd, with a vulgar lorgnette in her hand, filled his whole life now, was his sorrow and his joy, the one happiness that he now desired for himself, and to the sounds of the inferior orchestra, of the wretched provincial violins, he thought how lovely she was. (3.26)
After months of judgment and superiority complexes, Gurov admits that he judges Anna as below his class – and that he loves her anyway.
She glanced at him and turned pale, then glanced again with horror, unable to believe her eyes, and tightly gripped the fan and the lorgnette in her hands, evidently struggling with herself not to faint. Both were silent. She was sitting, he was standing, frightened by her confusion and not venturing to sit down beside her. The violins and the flute began tuning up. He felt suddenly frightened; it seemed as though all the people in the boxes were looking at them. (3.30)
Notice that Gurov doesn't feel the shame of their affair until Anna reminds him of it.
And Anna Sergeyevna began coming to see him in Moscow. Once in two or three months she left S., telling her husband that she was going to consult a doctor about an internal complaint – and her husband believed her, and did not believe her. (4.1)
It's interesting how details like this – or the reaction of Gurov's wife, for example – are completely glossed over. It's as though they just don't matter to the two lovers.