My experience with Kotak Mahindra Bank's internet banking has been very good. SBI sucks. Not very user-friendly,cumbersome and absolutely next to nil customer service/helpline.I use their internet banking only to check my account balance.
In public banks i will suggest PNB (min deposit just Rs.1000/-) you get a debit card & online banking.
It has a good online network (Core banking supported with tulip connection)
Most of the transactions (be it online transactions such as bills, neft/rtg etc work very well)
The site interface is a bit old (designed by infy) but works well.