There is a small difference in a normal saving account and a basic saving account. Earlier Public Sector Banks used to open No Frill Accounts with zero balance to increase financial inclusion in the country. In these accounts there was a limit of only four transactions a month and the KYC norms of the banks were not that stringent as required for normal saving bank account.
In 2013, No Frill Accounts were replaced with Basic Saving Account. No minimum balance is required to be maintained in the account. Also, no account keeping charges, ATM charges etc are levied in these accounts. However, there are certain restrictions with these type of accounts like, total credits can not exceed one lakh in a year, balance cannot exceed fifty thousand rupees at any time etc.
Since these accounts enjoy some above mentioned concessions on charges, you can not open more than one Basic Saving Account in a bank as per RBI guidelines. However, since these accounts are not traced and also RBI guidelines are silent on this particular question, one can surely open another Basic Saving Account in other bank.
It must be noted that one can't even open another normal saving account in the bank, with which he/she is having a Basic Saving Account.
Hope it helps.