How secure is it to connect an Arduino to the internet? Does it support any secure protocols?
A better option is something like the Arduino Yun which has two processors, a small Atmel microcontroller and an Atheros system on a chip running OpenWRT linux. The linux processor can act as a bridge to let the microcontroller communicate with the internet pretty easily.
This advice goes for many other set ups as well.
I use it together with a pcDuino and it works like a charm.