Unsecured network scaring

I'm so evil.

I was talking to my mom last night, telling her how I went to some dive bar the other night. She was being a mom, telling me how dangerous dive bars are. Meanwhile she was getting upset that she couldn't find her usual wireless network. My mom lives in an apartment with lots of unsecured wireless networks floating around. Why pay for internet when someone else is? She couldn't get online, so couldn't pay the bills she wanted to last night. So after the dive bar chiding, I chided her about logging into her banking site on an unsecured, foreign network. I told her they could easily log all the traffic back and forth and get all her passwords and other information. She said, "If they're too stupid to not turn on their network security, they're too stupid to do all that." I told her that's what they want you to think so you'll use their network while they gather login information, credit cards, and other private goodies. She thanked me that she'd be worrying about that all night now.

But thinking about that made me realize how little I know about network security. If the information was sent as clear text over an unsecured, http connection, I'm sure a router could be programmed such that it could scoop out your information. But will an https secured connection prevent that? I don't know. I should know that...