Seattle to Vancouver
When arriving at the train station and checking-in your passport and documents will be checked. Your documents will also be checked when arriving at Vancouver.
The Amtrak website provides some guidance for this:
The form of identification you carry with you on your trip should be the same piece of ID that you selected and provided details about when you made your reservation.
You must carry the original, valid identification document. Copies, expired or otherwise invalid documents will not be accepted. Each passenger must have his or her own ID.
It is recommended to arrive 1 hour before departure.
Amtrak Cascades from Seattle to Vancouver - What is the border crossing like northbound? summarises this nicely:
For reference this train does Canada Customs clearance on arrival at Pacific Central Station in Vancouver, not at the border.
Vancouver to Seattle
There will be a check at the station in Vancouver and another customs check when the train arrives in the USA.
It is also recommended to arrive 1 hour before departure.