Why did the professors insist that the vote on Elizabeth Blackwell's admission be unanimous?

The admission of Blackwell was unanimous because the students vote yes and because the professors thought that a majority of students will be against that a girl will come to a Medical School.

Actually, it wasn't the professors who voted-- they arranged it so that the students (all of whom were male) would vote; the assumption was that most or even all the students would vote against having her in their class. Elizabeth Blackwell was accustomed to rejection by this time-- she had applied to more than fifteen medical schools, none of which would give her a chance. She then applied to a small western New York school, Geneva Medical College in 1847. The faculty evidently didn't want to be seen as intolerant, and their clever plan was to put everything onto the students. If even one student said "no," that could be used as "proof" that her presence would cause problems. The school could then claim that was why she wasn't admitted. But for whatever reason (some say it was just as a prank, others say they assumed she would never actually show up for classes), the male students unanimously voted "yes." To everyone's shock, she accepted and showed up ready to begin her studies. And she graduated in 1849, the first woman in the US to get a medical degree.

There were no women medical students before her. If any professor objected to her, he would have been able to cause her a great deal of trouble and perhaps fail her unfairly or make her so unhappy that she would have quit school.