should I try to let the owner know via email that the router is wide open?
I would, if I knew them well enough and my neighbors and I had a civil relationship, which my family doesn't get along with our neighbors that well here in my house, so I'm sad to say that if that happens to my neighbors, no one in my house would care. At least we sleep peacefully knowing we've got our own wifi connection or whatever, our own service.
Personally, I'd try to notify, but wouldn't push.
Exactly, it all depends on the relationship you have with your neighbors.
Must be a pretty strong wifi signal. Mine isn't good beyound our front yard...

I'm new here and I had to find a laughing smiley because that was funny. Our wifi signal is not that strong either. We have four different devices connected to the internet 24/7 in our house, and I noticed yesterday how sometimes I have trouble web browsing when the other three computers are connected to the net doing random things.
I had very earnest young neighbours warn me that I'd left my wifi connection open.
They were thoroughly confused when I explained that I did it deliberately - I don't get charged for extra bandwidth (they just throttle it) and I don't use the entire amount .. why shouldn't I give it away to my neighbours?
To keep them happy I put a lock on it. And changed the name to 'Public Wifi: Password=password'.
I know people use it. That's fine with me.
Mac
At the last house I lived in, I would share my internet connection, and it was secured. Yes, I gave away my password because those neighbors didn't have their own internet access and I noticed I wouldn't get charged for extra bandwith, so I would let them use it. No problem. If this internet service were under my name and I got along well with my neighbors, I would do the same thing here, too.
I wouldn't unless you know them fairly well.
It freaks some people out--a did this for a neighbor and gave them a print out of how to change it and why they should (you are legally culpable if your WiFi is used to access illegal files).
They were deeply suspicious.
Yes, you have to be careful because sometimes when you try to do something for the own good of the other person, they become suspicious of your true intentions and they hold you responsible. It's like they're saying, "How the hell do you know, because you're using my internet connection, too? Why don't you get your own?" You might want to think twice about this. When we get services like that one we have the option to put a password on it to secure it. Just think about it this way: if they have no password it's because they don't mind other people using their internet. If you don't get along with them that well or they just don't trust you, leave it alone. When it does bother them for other people to use their service, they'll put a password on it to secure it.