Ms Rantzen was due to launch a fundraising campaign at the fifth birthday of the Childline Yorkshire and North East centre in Leeds on Wednesday.
Over the past five years the centre has provided more than 74,000 children from across the region with counselling and thousands more have been given information and advice.
But Ms Rantzen said more cash was needed to fund extra counsellors at the Leeds centre.
"Although the opening of the centre in Leeds has provided a lifeline for many children and young people who may have had nowhere else to turn, demand is still so high.
"Less than half the children who call can get through - simply because of lack of funds."
Recent figures show that the situation in Leeds reflects a similar picture across the UK.
Around 4,000 children call Childline every day, but lack of funds means that only 1,800 of them will get through to a counsellor.
Common problem
Childline Yorkshire and North East director Norman Titus said it cost £1,375 to recruit and train each volunteer counsellor.
"We need at least 80 times this amount in order to meet the demand on our service from children throughout Yorkshire and the North East," he said.
"We know that children who pluck up all their courage to call Childline, only to be told all the lines are busy, feel terribly let down."
The most common problem children call the charity about is bullying.