I just got Verizon DSL service and I was wondering for security if it's safer to disconnnect from the internet (but keeping the router/modem on so that just the DSL LED is on) when I'm not using it to protect my computer from hackers and to protect the router/modem from being used by unauthorized people. I already set the router/modem to not broadcast the SSID (maybe disconnecting the internet would be overkill after that?) and I changed the Firewall settings to medium.
Just lock your router's Wireless end down with a WPA2-AES encrypted connection as the security (change from WEP or WPA if need be) and use a strong passkey on the Wireless network. SSIDs can still be sniffed out of the air but many people are unlikely to do that. The router itself has a Backdoor which allows Verizon to access it whenever the modem shows it is online (could be at boot, could be overnight, whenever) but otherwise, if you're afriad that there may be a risk of leaving the router on, I suppose it doesn't hurt to disconnect it. But, either way I would say you don't much to worry about. There isn't much for them to get, and if your computer is off it shouldn't be accessible. Wake on LAN is the exception but the router won't pass such packets in a general stance without being set up to do so.
I've had my modem and router on and connected for years and have ran all sorts of applications on the connection. There isn't much to worry about.