Birmingham and Denver are about 1,093 miles apart by road. Split the difference and you arrive near Oklahoma City, the city closest to the halfway point between them. That puts roughly 13 hr 5 min of driving on the Birmingham side and 10 hr 34 min on the Denver side — the fairest single meeting point among the cities near the middle.
Over this distance most people will fly rather than drive the whole way. Oklahoma City still makes a fair, central place for Birmingham and Denver to converge, splitting the travel instead of asking one side to cross the country.
If Oklahoma City doesn't have what you're after, Kansas City and Dallas are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Birmingham and Denver.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.