Are you planning to travel the whole route of the Empire Builder? In coach or in sleeper? If you're in sleeper there's nothing to worry about. The train will certainly be less crowded than it is in the summer, and all of the services will be available.
If you're in coach, I'd expect that things will be pretty crowded right before and after Christmas, especially on the eastern part of the route, say Grand Forks to Chicago, in both directions. By New Years, I'd expect it wouldn't be so bad, but it depends on how the weather is, and also what time of the week the holiday is. I am often on the Empire Builder from St. Paul to Minot at Christmas time, and often when I'm in coach I wish I'd sprung for a roomette.
One thing to consider: as JayPea mentions, nights are very long on the Hi Line in December and January, and the much of the daylight you do have will be spent on the prairies of Eastern Montana and North Dakota. To my surprise, some people find stubble fields and snow to the horizon to be a bleak and depressing landscape. To me it has always meant that I'm almost home.
If I had to choose a month to take the Empire Builder, I'd probably go for late May. The prairie should still be green, and you'll have lots of snow in the mountains, and lots of daylight to see it all.
I think the weather-related delays are more common later in January and February, and with the floods in the spring, but that could just be my imagination.