Showing me websites you like gives us a common ground to begin our discussion of what you hope to achieve. We can talk about what you like best about another site. Is it the colors that capture your attention? Is there a particular feature you find appealing?
Sometimes, however, you just can’t get there from here.
Is there an easy way to customize the Genesis News Pro theme to make it look like the New York Times website? Any and all suggestions are welcome.
Sadly, I think that would be very hard for you to do.
Even though the News Pro theme is great for a news layout, the New York Times website would be a custom built theme of far greater complexity regarding widget/content areas.
You would be able to replicate some of the font styles and sizes, colors, etc. to give a similar overall typographical texture, but I think replicating the grid layout would take a lot of work from an advanced developer.