Are there any simple steps I can follow to find that out? Or should I simply not bother and get a new one?
There is one simple test you can run to see if the FTDI USB serial converter chip is functioning or not. Remove the AVR chip from it's socket. Wire a jumper from pin 0 to pin 1. Then plug your board into the PC's USB port. If the PC 'sees' the connection that is a good sign, if not, it's probably toast. Now launch the Arduino IDE and select the serial monitor. Now type something in the monitor and send it and if you get the same characters back then the FTDI device is probably OK and you can then just purchase a replacement AVR chip (with bootloader) for around $6-7 and be back in business. If the FTDI chip does not seem to function then the board is probably a write off unless you have experience unsoldering and soldering in a new FTDI SMD chip.