Why do the doctors blackmail the handmaids?

The reason is never explicitly stated; however, we are told on page 205, “Sometimes they blackmail." We are also told that this is how Janine (Ofwaren) was able to have her baby and that her commander's wife was aware of the arrangement.
On one occasion Offred visits the doctor for her regular exam, and the doctor offers to impregnate her, stating that most of the commanders are too old or sterile to produce a child. Offred politely refuses his offer while thinking,

He could fake the tests, report me for cancer, for infertility, have me shipped off to the Colonies, with the Unwomen. None of this has been said, but the knowledge of his power hangs nevertheless in the air as he pats my thigh, withdrew himself behind the hanging sheet.

So, with this textual evidence in mind, we can infer that the doctors have blackmailed in the past or are capable of doing it in the future because of the power that they have. They would do this in order to be able to have relations with the handmaids, which does not necessarily mean that they are sympathetic to the women's problems, but instead, they are willing to abuse the power of their position.