Was there negligence?
Whether the mother was criminally negligent is being examined by the Cincinnati Police Department.
On the other hand, many are asking if the zoo was negligent by not making the gorilla enclosure inaccessible. The likely standard the zoo will be judged by in a civil lawsuit, if one is brought by the parents, is whether the zoo had knowledge of the defect and failed to take reasonable precautions to protect its guests. Two years ago, the Pittsburgh Zoo settled a lawsuit with parents who lost their two year old child when he was mauled to death by endangered painted dogs after he fell into their enclosure. The child's mother lifted him up over a 4 foot wooden railing so he could better see into the enclosure. He lunged forward and slipped from his mother's grasp. He bounced off a net meant to catch debris and trash and fell into the exhibit. The zoo claimed that the mother was solely responsible for the loss of her son. The parents claimed that the zoo was aware of the defect in the exhibit - and produced safety committee meeting minutes from five years before the tragic accident - observing that on four occasions, parents had been observed dangling their children over the enclosure. The settlement was confidential.
Like the Pittsburgh case, the Cincinnati Zoo could argue that their liability, if any, was the result of the parent's negligence. If the doctrine of comparative fault is available, the zoo's damages can be reduced by the amount of the parent's fault. For example, if a jury decides that the zoo is 25% at fault and the parents were 75% at fault, and the jury awards the family $5,000 in damages, the zoo would only be responsible for $1,250.
A major difference between the two cases: in Harambe's case, the child suffered only a minor concussion from falling into the moat. In the Pittsburgh case, the child died. The amount of legal damages is vastly different.
Some states have laws making the owners of wild animals, including zoos, strictly liable (a guest is hurt by an animal, the zoo is automatically responsible for their damages). I am not aware whether this is the standard in Ohio.