HDFC seems to be rock solid w.r.t to least down times and the best Internet banking experience. Can't say about Phone/Chat support as i hardly use it. Axis used to be second till 3 months back but i would place ICICI
as second before AXIS now. Since Axis has unified their Internet banking, there have been lot of downtimes, the UI is messy, and more of browser dependent. (loves Chrome, Hates edge). ICICI has all the things in place but given the sheer number of accounts they have, it starts behaving like SBI during peak times. HDFC still gives kind of premium feel.
On a side note, If you have Citibank available at your place, then it's the best out of all these.