Not sure if you're being sarcastic or not but I've had lots of sites ask me for this, it's a way to verify your bank account is yours.
If you feel safer doing the 2 deposit thing then I guess just do that.
I do a ton of online banking and I can promise you other then spam emails from Nigeria I have never had anyone ever ask me for my username and password of my bank account.
This is 100% not normal and I would never ever give that info to anyone ever.
Heck even CHASE bank itself does not even know my username and password.
Its encrypted in there server and no human that works at CHASE knows this except me and Ms Chop.
Giving that info to anyone is a 100% no no.
Just do the 2 deposit thing.
Thats all thats needed.