This thread interested me, with my professional hat on. Info security is my business for big banks, airlines, govt etc. There is so much paranoia about making wi-fi networks security (in a nutshell: they ain't - businesses use a VPN to have a secure transport layer - but I digress).
To hear of neighbours who have open networks, and then freely and knowingly allow connections is staggering! Many may be simply for home use, but how many of these people are working from home with their corporate laptop connecting into the local wi-fi hot spot?
Regarding nefarious downloads - the authorities would have to demonstrate where the target data ended up. If the ISP demonstrated that a traced IP connection ended at your access point / router then that would give them a lead to your house and you might get your kit confiscated while it is scanned. At that point you might be wise to 'fess up that you also have an open and shared access point so that it may be monitored to point to the real purpitrator and get you off the hook. I'm not certain about the details of the law regarding computing devices used for transfering data in criminal acts but it can't be too onerous otherwise every router on the internet backbone would be confiscated.
To 'hide' your home network you can take some simple steps: Don't broadcast the SSID (the network name), restrict access to known MAC addresses (the lowlevel hardware address of your network cards), use encryption and a shared key between the router / access point and the laptop / pc. Make the key as long and cryptic as possible. Log access. If possible physically switch off the wlan when it isn't needed.
This is like securing your house - it isn't perfect and a determined attacker will get through - but in most cases they will move on to the house next door.