If someone latches onto your open network, they would be able to see your File and Printer Sharing. Then if you have no passwords on your file shares, they will be able to copy out all your shared files. They might even plant trojans into your file shares, waiting for you to click on them. Worms also make use of file sharing to spread. Trojan and worms belong to the realm of hacking, but hacking tools and tutorials are freely available on the internet, and one does not really need technical skills to follow a tutorial or use those tools.
The access point router may have logs. Mine does. So depending on what you want logged, you might see a history of ip addresses you accessed. The thing to do is to change the admin password for your router. Make it long and complex.