There are different kinds of LED strips... Does it light-up when you connect 12V (and ground) without the transistor?
You can test the setup (with the transistor or MOSFET) with a regular LED (with a ~1K current-limiting resistor). And if it's not working, you can check the Arduino's digital output with an LED (and resistor). Do you have a multimeter?
...If the transistor overheats and dies, it will usually short-out so it will be stuck-on and won't shut-off.