Among the 3 you mentioned, IDBI bank is best now a days. It is a PSU/sarkari bank so it is 'safe'. In case of a banking crisis like the one that happened in 2008, your money in IDBI bank account will be backed by the Indian government. This is important because in last 10 years a number of Private banks have closed (e.g. Global Trust Bank, United Western Bank) leaving their customers in lurch and desperately trying to ge the money out of their accounts with troubled banks. A lot of services in IDBI bank are free without the need to maintain big minimum balance. You are charged for every small service by both PSU banks like SBI or Private banks like HDFC. Further, IDBI bank staff will never call you and attempt to sell you useless investment products like ULIPs. Also IDBI bank is offering best interest rates on fixed deposits now a days among all PSU banks.
Axis bank will charge you for things like cheque books, Pay orders, money transfers by NEFT, Debit cards and you will need to maintain a minimum balance. Further they will try to sell you insurance products like ULIPs, gold coins etc. from time to time. You will get tired of answering their phone calls every day. They will also want you to take their Platinum credit card. When I refused to take their credit card, they sent one of their executives to my home on a Sunday morning anyway, just to push the credit card down my throat. Needless to say, I soon closed my account with them.
HDFC bank staff will also try to sell you ULIPs. You will maintain a minimum balance and will pay for services that are free with IDBI bank. Further every time you receive a big payment in your account by cheque or by online transfer, they will call you and ask you to tell the source of funds. If you ask me, this is a big nuisance. I have a business to run and fielding useless calls from bank must not be a part of my daily routine.
If you have receive payments from abroad by wire transfers, then CitiBank business current account with minimum balanc of Rs 50,000 has lowest charges for receiving wire transfers. They have few branches but their service is better than the most banks in India. I don't know if they have a branch in your city and near your place.
HSBC Direct account with zero balance is also good if most of your business is done online and you don't need to deposit or withdraw cash from bank frequently. Have a look at their charges:
http://www.hsbc.co.in/1/2/business/business-current-accounts/hsbc-direct-for-business
http://www.hsbc.co.in/1/PA_1_083Q9FFKG80E20RA9Q00000000/content/website/pdf/business/bd_tariff.pdf
Once again I don't know if HSBC has a branch near your place. My office is located in Central Delhi (near connaught Place) and all banks have branches in Connaught Place. So all banks are accessible for me.
Given my bad experience, I suggest NOT opening accounts with ICICI or Standard Chartered Bank.