It depends.
* If you're texturing it with a cube-map or a procedural function, then they both work, but the box is simpler.
* If you're texturing it with traditional UV mapped textures, then the dome may work better, and will give a better parallax effect.
* If your sky shader performs any kind of per-vertex processing that's interpolated, it may work better on a more detailed mesh, such as a dome.
* If it's possible to see out the bottom of the dome, you'd have to cap the bottom of it.

Each has its uses. Which is better depends on many things.

Personally, I like the way skydome looks when the radius is close to the curvature of the real life environment. For Earth, the curvature is about 8 inches every mile, so if the designer wants to convey a realistic gameplay environment, then this needs to be at least considered. An exception is when the game is meant to look surreal or mythological, then the design tends to usually look good with a smaller curvature.

Skybox can often be easier to create and looks close enough to a real life planet environment that many game designers use it. If artistic liberty is available, any small inconsistancies can be hidden with mountains, city buidings, and sometimes vegetation.

On the other hand, I have seen very non-natural looking skyboxes and skydomes used to portray a dreamlike state for the game player, sometimes not even intended to model sky but a fantasy environment. Once I saw a demo of a game which had animations on the skybox or skydome, so this shows the flexibility with them.

As for creating a bottom, as Hodgman well suggested to consider, some games have a sea, desert, or other. The rest of the terrain mesh can emerge through this bottom like an island or oasis.

Clinton