When I put a graph on a layout in Prism 4, its background color shows as a border around the graph.

When you first put a graph onto a Prism layout, that layout gets the same background color as the graph. You can change this using Change...background color.

With Prism 4.00, sometimes changing the layout background color doesn't entirely work. The graph itself still has the background color of the graph sheet. This is due to a bug in Prism 4.00, but one that is easy to work around. It only happens when the last time you exported a graph you chose to include the background color in the exported file. To solve the problem, therefore, use File..Export to export a graph (any graph) and for background color choose "Make clear". Now the layout will work as it should.