This is probably the easiest winter event I've seen in all the MMOs I've played. Dungeons and Dragons Online's race involves a set of ski-jumps onto hovering and slippery ramps. You fall off with one wrong move and have to re-start the entire thing. That thing took me hundreds of runs just to master. STO's run is much easier in comparison.
1) Make sure mouse view and auto-run are enabled. This allows you to do the course simply with little mouse movements.
2) Find long straight or slightly curved paths for fast runs
3) Make use of snow banks and humps to slow down in order to turn fast around sharp bends
4) Practice, practice, practice
After around 10-15 runs, you should be able to consistently complete the entire run.