How to prevent orphaned table headers in Microsoft Word?

Highlight all the content inside the header row and the second row and go to Paragraph properties and check Keep with next. This should stop the header row and the second row from being on separate pages, while it still allows the rest of the table to freely break across pages.

Per @Adam's answer, the "Keep With Next" option needs to be set on the header paragraphs.

I would recommend setting up a "Row Header" style that encompasses this paragraph option, as well as any other visual formatting, that can be easily applied to the top row of all your tables.