Jie, this is dependent on desired application requirements. As for dimension lowering the density of states become higher at the lowest available states and some scattering transitions becomes prohibited due to quantum confinement, that can be useful for electron transport properties and for long exciton lifetimes. For photovoltaics also a spatial topology of the active area is important, because for 1D wires the absorption probability higer as the multiple reflection from the surface in turn guides the light to the neighboored wire. Also in the 1D radial heterostructure quantum wire one can obtain spatially separated transport of photoexcited electrons and holes so their recombination is strongly suppressed while they drifted to the contacts. All meantioned approaches indeed enhanse the quantum efficiency.

"Better" needs to be defined... in practical devices, one is concerned with amount of charge/current transported for example. If the highest mobility comes with a severe limitation on amount of current that can be supported, is that a "good" conductor?

For some applications 1D is advantageous and some other cases 2D is better. It depends on case-by-case. But if you want to compare 1D & 2D effects side-by-side in any particular case, you can certainly do that. In that case you should choose similar material composition for 1D & 2D materials and compare them in similar application. Here is one of my previous review paper where I was exactly trying to do that. This was the comparison between Carbon 1D material (CNT) and Carbon 2D material (Graphene) in electronic and optoelectronic applications. If you are interested you can check that out..

Article Graphene Versus Carbon Nanotubes in Electronic Devices