I see a couple problems.
1) You need to make sure there is a ground connection from the breadboard to the Arduino board or there is no common referece for the circuit. (as MarkT mentioned)
2) For a proper transistor circuit, there should be a leakage drain resistor from the base to ground. Try a 470 ohm (1/4 watt). Usually transistors are good enough these days to do without it, but good circuits are more reliable.
To test whether the transistor is shorted, take the line going to the Arduino and and uplug it. If the LED is still on, then the transistor may be bad or very leaky. Also ground that same line. If the LED is still on, then the transistor is definitely bad.
Or you could just measure the voltage on the line to the arduino. That would tell you if it's an Arduino issue or transistor issue. But definitely connect a ground line, and make sure you have a 1/2 watt 470 ohm resistor for that LED.
One last thing. It looks like you're using a 470 ohm transistor on the base circuit. That means the Arduino will be driving about 10ma at the max. If that's the case, then why not just drive the LED directly with the Arduino (as Crossroads suggested)? You probabbly want more like a 4.7K resistor there (not that this would cause your symptom).