I wouldn't worry too much about it if you're dealing with them completely online. Just transfer your money in, check your online statement once in a while, then transfer out when needed.
I recently documented the sign-up process with ICICI in a series of articles (see the Articles page) but haven't had a chance to test out their website interface much yet. The sign-up process was rather uncoordinated, but everything worked out fine, without me needing to contact them at all. They seem to be legit, but I'll just try and avoid talking to them.
For the most peace of mind, I've found Citizens Bank (even though they have no branches) to have the best service. This is perhaps because they're a Vancity initiative…
Citizens Bank is my favourite bank hands down. Their current rate is 3.8% (you'll find they're always 0.05% higher than ING).