Here's an article about the internal checkpoints.
en.wikipedia.org
According to the law referenced there:
"Under US law, people who enter the US without inspection (EWI) can be subjected to expedited removal if they are found within 100 miles of the border."
This suggests that either permanent or roving border checkpoints can only be conducted within 100 miles of the border. I'm not sure whether the law considers to coast to be the "border," but even if it does, most of the route of the California Zephyr is located more than 100 miles from the border. Thus, I don't think the Border Patrol does inspections on that train.
However, it's been reported here that many other local, state and Federal law enforcement agencies occasionally do searches on the California Zephyr, mostly for drugs, but also perhaps other contraband, or they're looking for a particular person. When you buy an Amtrak ticket, you agree to have ID available for inspection if requested, and I would imagine that if a cop questions a person and has "reasonable suspicion" that the person might not be documented properly, they can ask for ID or other documentation. While some law enforcement agencies don't care about a person's immigration status, other do care, so one still needs to be careful wherever one goes.