The association said in a statement on Sunday that the decision to reopen "was taken after consideration of the current difficult security conditions and the need to maintain the safety of customers and employees alike, in the absence of adequate protection by the state".
It added each bank would determine its own channels for banking operations with commercial and educational institutions, and the health care sector amongst others.
A top Lebanese banker on Friday criticised politicians for failing to enact a capital control law, saying this was the way to avoid bank raids by savers demanding funds from frozen accounts and to stop banks' "discretionary practices".
The holdups reflect savers' desperation three years after Lebanon's financial system collapsed due to decades of state corruption and waste, and unsustainable financial policies.
The government has agreed neither a financial recovery plan nor enacted reforms deemed vital to get Lebanon out of the crisis. While the government says it is committed to reforms, the International Monetary Fund says progress remains very slow.
(Reporting by Timour Azhari; Writing by Moataz Mohamed; Editing by David Clarke, Alexandra Hudson)