Many families are in an "extremely precarious" financial position, leading them to take out loans to pay for necessities, the Children's Society and StepChange debt charity said.
The stress of keeping up with repayments leads to rows, emotional distress to children and cutting back on essentials, the charities said.
A survey for the report found "problem" debt - defined as being in arrears on at least one household bill or credit commitment - currently affects close to one in five, or 18%, of households with children in the UK. On average these households owe £3437 each.
The findings amount to 1.4 million families in "problem debt" across the UK, the charities said, and a further 2.9 million are struggling to keep up with payments.
The Government should work with creditors and other groups to develop a "breathing space" scheme to give struggling families an extended period of protection from default charges and enforcement action, the report said.
Matthew Reed, Children's Society chief executive, said: "Families are increasingly relying on debt as a way to make ends meet - but we're in danger of ignoring the impact this is having on children now and in the future. We cannot allow children to pay the price of debt."
Mike O'Connor, chief executive of StepChange, said: "This report is a stark warning to policymakers, creditors and the wider society of the devastating effects of debt on children."