Is it safe to authorize Coinbase to withdraw cash directly from my bank account? I wanted to buy Bitcoin. Coinbase said that they could transfer money directly from my U.S. bank account and convert it to Bitcoin, and then I'd have a Bitcoin address and could give people Bitcoin or pay with it for things. I went ahead and signed up and requested a transfer of $100. 

Here's why I am worried: Coinbase asked me to provide my credentials to my online banking while I was on the Coinbase website, both my username and my password. I thought that should have happened on my bank's website with SSL connection, not through Coinbase. I went ahead and did it but now I'm worried. Coinbase said that the transaction wouldn't go through until January 6, 2016 because of ACH and the holidays, which is fine. I'm just worried because I have several paychecks worth of money in that bank account.

(I did not access Coinbase from Android or any other mobile device. I used my desktop PC with Google Chrome web browser.)
They don't  need your password to process the ACH in the future. this sounds like they used your banking credentials to verify that it really is your account. You should be able to change your password on your banking account now and be safer.