Mrs. Putnam is a minor character in The Crucible and we do not know much about her. Miller describes her much more briefly than her husband and most of the other characters, saying that she is "a twisted soul of forty-five, a death-ridden woman, haunted by dreams." Her questions to Mr. Parris in Act I immediately establish her as unusually credulous and superstitious, even by the standards of seventeenth-century Salem. It is clear also that she has been devastated, perhaps driven somewhat mad by the loss of so many children.
Mrs. Putnam's primary motivation, therefore, would seem to be vengeance. She is superstitious enough to believe that Rebecca Nurse used witchcraft to murder her children and, more generally, that the devil is abroad in Salem and all those accused are guilty. Her mania may mean that she is not even particularly interested in the guilt or innocence of those accused, as long as someone suffers for what she has endured.
Thomas Putnam, of course, uses the hysteria as a way of enriching himself by appropriating land from the accused. Mrs. Putnam may share this motivation, but it seems to be secondary as there is no direct evidence of it in her words.
To a large extent, Mrs. Putnam has been unhinged by the loss of so many of her children. She desperately seeks an answer as to why her children keep dying in infancy. Infant mortality was shockingly high in those days, but without adequate medical knowledge, Mrs. Putnam starts looking for a supernatural explanation for her tragic losses. It is for that reason that she is so inclined to ascribe the loss of her children to witchcraft.
That said, Mrs. Putnam is still sane enough to realize that the witch-craze presents great opportunities for personal enrichment. Once would-be witches have been arraigned by the authorities, their land becomes available at a knock-down price, and Mrs. Putnam, like her equally greedy husband, can't wait to get her hands on it. One could say that in this unseemly land-grab, Mrs. Putnam is in some way compensating for the loss of her children.
Mrs. Putnam's character motivation comes from a variety of previously vengeful situations.
The narration at the beginning reports that she and her husband had suggested a new minister for the town, James Bayley, who happened to be Ruth's brother-in-law. The majority of the town agreed he should be hired, but a small group made sure he did not get hired. The reason for this is never cleared up in The Crucible. For Ruth's sake, this certainly gives her reason to have anger against that small faction. Many people can accept a 'no' answer when a justification is provided; but when it is unexplained, anxiety surfaces. That anxiety can quickly turn to anger or vengence.
Mrs. Putnam also seemed particularly put off by her first encounter in the play with Mrs. Nurse. Rebecca Nurse called into account her expertise as a grandmother and mother while knowing that Rebecca has lost several children. Although we find Rebecca a woman of formidable character, Ruth Putnam could have taken the comment as if Rebecca was prideful.
The Crucible never states if this group that stopped the ministry of Bayley included Rebecca Nurse, but leaving the group vague allows the mind of the reader jump to that conclusion.
Mrs. Putnam further struggles with both the Proctors and Nurses because of a land dispute her husband has with both Francis Nurse and John Proctor.
Ruth Putnam's motivation is vengeful because she feels her family has been attacked and her ability to raise children has been questioned.