This AP article via Boston.com
provides a brief overview of the online checking accounts offered by Bank of America, Ally Bank and ING Direct. You might ask why BofA is in this group. It's due to their new eBanking checking account which waives monthly fees if only bank online and at ATMs (avoiding bank tellers). The article implies that rare branch visits for things like depositing coins or resolving complicated customer service issues would not result in a monthly fee. I wasn't able to confirm this via Bank of America's online documentation. Here's an excerpt from BofA's eBanking disclosures
You avoid the monthly maintenance fee for each statement cycle during which you (1) use only self–service options for your deposits and withdrawals and (2) for which we provide you with only a paperless statement.
(1) Self–service options mean that you conduct the transaction yourself, generally by using one of our electronic services, and that a teller or other bank representative does not help complete the transaction.
If you want to avoid monthly fees, you would be much better off with the other two banks with online checking accounts mentioned in the article: Ally Bank and ING Direct. In my opinion, Ally Bank's Interest Checking is the best since it allows paper checks (that are free) and offers full reimbursements of ATM surcharges. Here's my list of the best internet checking accounts
based primarily on interest rates.