How does the pull down resistor work in this circuit?
I've seen some comments here stating that the pulldown is needed to keep the transistor turned off, instead of floating, or for noise reasons.  The base pulldown does in fact help keep the transistor off, but no one has answered why. This is why I chose to answer, rather than extend the comments. User alexan_e is thinking correctly about it, so I'll elaborate here.  

There is a Miller capacitance between the collector and the base of all BJTs. Most designers know all too well about the MOSFET's Miller Capacitance and forget that the BJT has some as well.  The BJTs Miller capacitance provides a leakage path from collector to base, injecting charge carriers into the base region, which can be amplified by the BJT's Hfe (gain). This allows noise current to flow from collector to emitter.  The inclusion of the base pulldown will provide a path to ground to discharge the Miller capacitance and keep the BJT hard off and noise free.