Dadaab is a collection of three camps, Ifo, Hagadera and Dagahaley, in northeast Kenya, approximately 80km from the Kenyan-Somali border.
The number of people there has swollen to nearly 400,000 at the settlement which was built 20 years ago to house 90,000 refugees.
Around 1,300 people more are arriving from drought-stricken Somalia and the Sudan every day, of which Save The Children estimates 800 are children.
The charity said the number of unaccompanied children arriving from Somalia has seen a four-fold increase and a small number of babies abandoned by their parents after arriving at the camps.
In July, Save the Children helped 80 separated and unaccompanied children who arrived at the camp from Somalia, up from a monthly average of 22 in 2010.
A spokeswoman said it is not clear why the babies were left, but staff say it is likely to be as a result of the desperation felt by parents who have often walked for weeks without food, water or shelter to reach the camps.
The charity has placed record numbers of unaccompanied children with foster families in recent months.
Among the children abandoned by their desperate families are two-month-old Nasib who was found by a roadside by a woman who took him to a police station.
She was asked by police to care for Nasib while they investigated the case and Save the Children is working with police to trace Nasib’s parents.
Another baby, Naifin, was left at the home of a registered refugee by his mother who had just arrived from Somalia.
She asked to stay overnight after arriving at the camp but fled in the night, leaving Naifin behind.
Prasant Naik, Save the Children’s Kenya country director, said: “We can only imagine the levels of desperation driving parents to abandon tiny babies in this way.
“After walking from Somalia without food and water, it isclear that some parents feel that they cannot care for their children any more, despite having reached help.
“Meanwhile, we know that some parents are sending their children to Dadaab alone, and other families are being split up as they make the journey.
“There could be no starker indication of how dire the situation for people caught up in this crisis has become.”
More than 40,000 people arrived at Dadaab from Somalia in July, the highest number of new arrivals in the camp’s 20 year history.