The United Nations Children's Fund (Unicef) said the floods were causing "havoc" and "chaos" in the region, with around 20 million affected and could be the worst in living memory in some areas.
The challenge was most acute in India's impoverished eastern state of Bihar where a number of pregnant women in flooded areas have given birth to stillborn babies, as flooding led to the collapse of the rural medical infrastructure in many areas.
On Friday, Surendra Srivastava, a police spokesman in Lucknow, the capital of the north Indian state of Uttar Pradesh, said: "At least 12 people, mostly children and women, died of rain-related incidents in the last 24 hours."
With rivers bursting their banks along the fertile plains south of the Himalayas, India has ordered the army to help evacuate people from some of the worst-hit areas.
Soldiers began to evacuate people from 500 villages under water in Uttar Pradesh on Friday, Umesh Sinha, the state relief commissioner, said.
About 100,000 displaced people were staying in government relief camps in Assam in eastern India, while hundreds of thousands more sought shelter on higher ground, setting up makeshift dwellings.
The floods have destroyed crops worth hundreds of thousands of dollars in Uttar Pradesh, Diwakar Tripathi, a senior government official, said on Thursday.
Medical teams were trying to visit different regions by boat to make sure there were no outbreaks of waterborne diseases like cholera.
In Bangladesh, 11 people died in flood-related incidents overnight as new areas were flooded - including low-lying areas of the capital, Dhaka.
More than half of the low-lying and riverine nation is flooded, with nearly seven million homeless or marooned.
The monsoon season in South Asia runs from June to September and is vital to the region's agriculture.
But the monsoons are always dangerous; last year more than 1,000 people died, most by drowning, landslides, house collapses or electrocution.