Bignose is on to something when he mentions vectors. Technically, the two vectors (Train car and rain drop(s)) are not in the same direction.
Rain pushes south, so to speak, while the train pushes east.
This gives you a final vector pointing south east. Now, the train is on a frictionless track, so it cannot go downward. The added down vector by the rain is canceled by the upward force of the car upon it, which has a now greater upward force from the track.
This leads me to reason out (although i can hardly believe it) that the train never changes x velocity, because there are no forces present to change x velocity. This would be very cool to test. : )
The problem is somewhat similar to a bowling ball rolling out of a plane, with no wind resistance. No matter how fast the ball drops, it still travels with a constant speed horizontally.
Also, i am assuming that the train car is not running into rain drops, because that would qualify in my book as a form of air resistance. The car only increases mass.