In The Crucible who does Mrs. Putnam blame for the death of three of her children?

I am glad that you put in the part about "three of her children."  The specificity is good, because the answer changes with that added information.  

Ann Putnam has lost seven total children.  She has one surviving child.  That child is Ruth.  Rebecca Nurse was Ann Putnam's midwife for all of those births, so Ann Putnam is a bit suspicious of Rebecca Nurse.  Putnam is also slightly resentful of Nurse, because Rebecca Nurse has...

I am glad that you put in the part about "three of her children."  The specificity is good, because the answer changes with that added information.  

Ann Putnam has lost seven total children.  She has one surviving child.  That child is Ruth.  Rebecca Nurse was Ann Putnam's midwife for all of those births, so Ann Putnam is a bit suspicious of Rebecca Nurse.  Putnam is also slightly resentful of Nurse, because Rebecca Nurse has successfully given birth to many children and has many grandchildren.  

However, Ann Putnam does openly lay the blame for three of her children's deaths on Goody Osburn.  Tituba is being interviewed (interrogated) about her involvement with the girls, and I believe that Tituba simply names some names to get the attention and blame off of herself.  She names Goody Osburn a witch, and Ann Putnam jumps on it as Gospel truth.  Putnam deeply desires to put blame on somebody for the deaths of her children, and Goody Osburn is as good as the next.  Goody Osburn even helped with three of Putnam's failed births.   

Mrs. Putnam: I knew it! Goody Osburn were midwife to me three times. I begged you, Thomas, did I not? I begged him not to call Osburn because I feared her. My babies always shriveled in her hands!