Summer camps for Jewish children started on Monday, despite parents' concerns over transport, security and food.
More than 50 families regularly require extra cash for food
In Thessaloniki, the country's second biggest city, community leaders warned that if salaries could not be paid, synagogues and other buildings may be forced to shut.
Directors of the welfare programme said the lack of available cash after banks closed meant it was impossible to continue payments. It is hoped the shortfall can be made up if the country regains a more stable financial footing.
Around 5,000 Jews live in Greece, mainly in Athens. They include many Holocaust survivors who do not use bank cards or electronic banking, meaning pensions and restitution payments cannot be paid to them during the current crisis.
The Jewish community has previously backed the country staying in the single currency and the European Union, but did not adopt an official position in last weekend's referendum.
Many younger Greek Jews have left the country to study or look for work elsewhere during the long economic crisis.