ASB reminded customers that its contact centre remained open but warned there could be delays getting through to someone.
"Please be patient if there is a delay as call volumes are much higher than normal."
The BNZ said it would close its branches and Partners Centres from Wellsford to Pukekohe on Monday.
"We hope to offer limited services in some branches across the region on Tuesday while we remain in Level 3, so we'll let you know as soon as we can."
Outside of the Auckland region, its branches would be open with level two guidelines, including physical distancing and other procedures in place.
A message on Kiwibank's Facebook page said all its Auckland branches would also be closed today and urged customers outside of Auckland to use internet banking, its mobile app, phone banking and ATMs before coming into a branch.
"If visiting us in branch, we ask that everyone does their bit to scan in and please stay home if you're feeling unwell."
ANZ's Facebook page said some selected branches in Auckland would be open from 9am to 4.30pm under alert level three guidelines. Eleven are listed on its website as being open in Auckland.
"To make sure we all stay safe, these branches will have screening processes, QR code check-in, physical distancing and PPE in place."
Outside of Auckland ANZ said its branches would stay open with social distancing in place.
Westpac said it would be opening 13 branches in Auckland from Monday to Wednesday.
"We ask customers to only visit these branches for urgent matters, or if they are unable to do their banking online or over the phone."
Westpac said all other branches around the country would be open with Covid-19 Level 2 protocols in place.
"Social distancing measures will be in place. It is important that customers do not visit our branches if they feel unwell."
Westpac said it invited customers to wear a mask while in its branches but said customers may be asked to remove them briefly to help with identification.
It also warned of possible high levels of demand on its call centre.
"We expect high levels of demand so please only call if you need urgent assistance."
The Government says it will put in place financial support for businesses if the level 3 lockdown goes on longer than three days.
People can go to the Ministry of Business, Innovation and Employment website to get approval to travel in and out of Auckland.