Some 87% of adults thought the system was "facing severe problems", rising to 94% among the over-55s, research commissioned by Christian think tank Theos found.
The majority, 68%, believe claimants should only benefit from state services if they have paid into the system, even if they are in real need, and the same amount said welfare should be a "safety net for only the poorest and neediest", according to the ComRes poll.
But half the population believe the wealthy should be able to receive benefits if they have paid taxes.
The findings comes as Prime Minister David Cameron was moved to defended the Government's welfare reforms in the face of criticism from the head of the Catholic Church in England and Wales.
Nearly six in 10 voters believed pensions, the NHS, tax credits, child benefit, disability living allowance and unemployment benefit, would be cut or have virtually disappeared within three decades, the poll found.
About one third believed politicians were "mainly to blame", while a fifth said false claimants were responsible, 16% blamed "benefits tourists" and 15% said the European Union was at fault.
Theos director Elizabeth Oldfield said: "We need to think carefully about what welfare is for."