Washington - The family of a toddler who fell into a gorilla exhibit in Ohio - facing a torrent of criticism after zookeepers were forced to shoot the rare animal - suggested on Wednesday well-wishers should donate to the Cincinnati Zoo.Images of the three-year-old boy trapped in the enclosure with the 400-pound silverback have resonated around the US and beyond, even reaching the White House press briefing room. In the statement, the family offered their thanks to zookeepers for saving their child and said he is doing well following "a checkup by his doctor.""Some have offered money to the family, which we do not want and will not accept. If anyone wishes to make a gift, we recommend a donation to the Cincinnati Zoo in Harambe's name," the statement said.The family has not been officially identified and a spokesperson said the parents had no plans to participate in interviews. White House Press Secretary Josh Earnest, asked about the gorilla incident, called it a "tragedy" and said it was not clear there was any role for federal officials to play in the aftermath.Criminal chargesThe Saturday incident, captured on smartphone video, has ignited a furious debate over the actions of zoo officials and the child's mother. She has alternately faced criticism and received support from well-wishers - with much of the debate taking place online. One petition on change.org is asking officials to hold the parents "accountable for the lack of supervision and negligence that caused Harambe to lose his life." The petition had more than 450 000 supporters by midday on Wednesday. The Cincinnati police say they are investigating the parents to see if criminal charges are warranted. Questions about securityThe zoo is facing scrutiny, as well. There has been criticism of zookeepers' decision to shoot the endangered animal instead of tranquilising it."Gorillas are gentle, nurturing animals who attack only when provoked, and experts report that Harambe was likely trying to protect this child," Brittany Peet of PETA, People for the Ethical Treatment of Animals, said in a statement.Asked to comment on Tuesday - as footage of the drama ran on a loop on US television networks - Donald Trump said it appeared officials had no choice but to shoot the gorilla. "I think it's a very tough call," said the Republican White House hopeful.Cellphone video of the incident showed the gorilla handling the child in an erratic manner - sometimes seemingly tenderly, other times roughly dragging him through knee-deep water. Zoo officials have said that they could not take the risk that the gorilla might hurt the child in agitation, if hit with tranquiliser darts."This is an animal (that) with one hand, I have seen take a coconut and crush it," zoo director Thane Maynard said.The zoo is also facing questions about the enclosure's security, since the toddler crawled through an exhibit barrier and fell into a moat."The failure of the Cincinnati Zoo to adequately construct this enclosure to protect both the public and the animal held prisoner there is a clear and fatal violation of the Animal Welfare Act," said Michael Budkie, the head of the organisation Stop Animal Exploitation Now, in a letter to the Department of Agriculture. The group, which claims the zoo has had other violations, wants federal authorities to investigate the facility. The boy was first identified as being four years old, but a family spokesperson confirms that he is three.