How to split two networks for one Internet connection

I have a single Internet connection and I'd like to create two networks to access it.

Network #1: WiFi accessible Network #2: Wired, not accessible from Network #1, but can still get to the Internet

Are there any appliances for managing this configuration?

You'll want to look into creating separate VLANs for the Wireless and Wired traffic. Separating the networks into VLANs will prevent the networks from being able to see each other at the data link layer. Your access point may already do this, but the procedures are different for each one, especially when using 3rd party firmware.

For example, DD-WRT, a 3rd party firmware originally for Linksys WRT54G series WAP/Routers can do this. See how to use VLANs to detach networks, or how to create multiple detached wireless VLANs.