He was addressing a congregation of 1,500 people at a special mass in Westminster Cathedral this week, to celebrate the 150th anniversary of the Catholic Poor-School Committee, set up to found Catholic schools.
Today there are 392 Catholic secondary schools and 1,854 primary schools, many of them popular and successful. However, like all schools, they are affected by the teacher shortage.
"We need more young Catholics to enter into the teaching profession," said Cardinal Hume.
"They need to be persuaded that few tasks are more noble and necessary than the Catholic formation of a young person for the responsibilities and opportunities of adult life.
"Clearly, the future of our Catholic schools depends on recruiting Catholic teachers and training them effectively.
"I do not, of course, forget that a very considerable contribution to Catholic education has been made by many non-Catholics who have shared our ideas and worked with us."
Cardinal Hume insisted that the Catechism of the Catholic Church must be "the basic text book" for Catholic schools, and called on parents to support the work of teachers. "It is tragic when what is taught in school is not echoed in the family."
In his introduction to the anniversary mass booklet Bishop David Konstant, chairman of the Catholic Education Service, writes: "In the 1840s public education was in its infancy. The few Catholic schools there were had been clandestine and only slowly came into the open in the wake of Catholic emancipation.
"Since those early, difficult days, we have relied for the development of our system of schools on unselfish giving of untold numbers of ordinary parishioners - we have depended literally on 'the pennies of the poor'. "
The anniversary comes at a time when the Catholic and Anglican Churches are concerned that Government plans for three new categories of school - "foundation", "aided" and "community" - will see the break up of the Church sector. Some church-controlled, aided schools, they fear, will seek independence through foundation status.