First, to be clear, I am definitely a proponent of some sort of rain shield and puddle mitigation here...
But to put all of this in accurate perspective: there is full overhead coverage for the three main doors of the South Station headhouse to the platform area -- it is visually quite clear, now that all construction barriers are down and the concourse is fully re-opened, that those are the main doors (i.e., beneath the train times ticker sign) and I am sure 90%+ of passengers are going to go that way. So this is basically a non-issue for most people, especially out of town visitors. It is only these doors on the far left and right "wings" where you've got this gap. SO, if you don't want to get rained on/step in puddles, just go through a door 20 feet to your left!