At the surface, this seems like a decent idea. Why not launch from higher up, so you don't have to go as far?
The answer is that when choosing a launch location, we are not so much concerned with height as we are with speed. To get into a Low Earth Orbit, you need to be traveling at about 7.8 km/s. At the equator, the Earth is spinning at 1670 km/hr, or ~0.5 km/s. This means that we need 7.3 km/s of delta-V
According to NASA, we can calculate the Earth's rotational velocity at any latitude using the formula $1670*\cos(\theta)$. Denver is at 39 degrees North, so the rotational velocity there is ~1298 km/hr, or ~0.36 km/s. So to launch from Denver we need to increase our speed by 7.44 km/s. This is a difference of 0.14 km/s, which is a significant saving.
Of course, we don't actually launch from the Equator, but we try to get as close to it as we can. All the different countries around the world try to launch as close to the equator as possible.
The other thing to consider is that the Earth is not perfectly spherical. It has an equatorial bulge formed by its rotation. This means that the radius at the equator is about 26 miles larger than at the poles. According to Wikipedia, this results in the equator actually being taller than Mount Everest. Of course, the atmosphere and ocean also bulges out like this, so you are still under the same amount of atmosphere, but it does make it easier to get to space. Since you're further from the center of the Earth, you experience more centrifugal force. This helps counter the force of gravity by a very tiny amount, but every little bit helps when trying to get to space.
As @OrganicMarble points out, it also allows us to drop our boosters in the ocean, to be retrieved later.