I tried to link my bank account with coinbase and they asked for my Login/Password for my online banking? It just felt wrong. The first thing people are told is not to give this info out. Is this something I should be ok with? I know coinbase is one of the biggest. Like I said it just felt wrong...
If you are uncomfortable with it, then don't do it.
I don't think that Coinbase has ever asked me to do this when adding a new a bank ccount. I think that the normal procedure is supposed to simply be send 2 small transactions or something along the lines of that. There are virtually no financial institutions, even Paypal, that ask you to directly enter the password of your bank details into them.
Perhaps that's been a long time ago, and things have changed since they've integrated into the traditional banking system. But I'm still skeptical and echo others who think that this may be a phishing site you're on.