There are two likely reasons that Mr. Easton may be embarrassed to encounter Miss Fairchild on the train.
First, Mr. Easton used to be one of Miss Fairchild's suitors. The text tells us that he once vied with an ambassador for the young lady's hand. So, meeting her on the train with his wrist handcuffed to an imposing man is rather embarrassing. Mr. Easton feels self-conscious because he knows that he is a criminal who is about to spend seven years at Leavenworth prison.
Seeing his discomfort, Mr. Easton's glum-looking partner takes pity on him and pretends to be the criminal instead.
There may also be another reason Mr. Easton is embarrassed to encounter Miss Fairchild on the train: he doesn't wish Miss Fairchild to alert common acquaintances in Washington about his criminal background. From Miss Fairchild's words about the "Washington crowd," it is likely that Mr. Easton was once a well-known figure there. So, he may not be too pleased about exposing his counterfeiting past to such a fashionable crowd.