The radiation from the tests tended to blow north and east. For example, see this map showing all paths where more than one mushroom cloud overlapped. And see this map showing Iodine-191 exposure by county.

So the majority of the radiation was deposited in midwestern states, and Utah, not New Mexico or the South-West.

Long-term exposure to low/medium levels of radiation (as opposed to high-levels, which can lead to acute problems like radiation sickness) increases the risk potential for various types of cancers and birth defects. It is a probabilistic, not absolute effect. So you can only see it when looking epidemiologically at a population — no individual cancer is usually traceable to a single exposure source, but aggregate cancers above the baseline can be attributed, especially certain types of cancers which are known to be linked to certain biochemical/radiological pathways (e.g. leukemia, which can be associated with "bone-seeking" fallout products like strontium-90).

So small populations make it much harder to detect a "signal" from the "noise." Over the aggregate, knowing how much radiation was put out, one can come up with gross estimates as to probable fatal cancers generated using statistical models. So, for example, it is estimated that US atmospheric nuclear testing caused between 70,000 to 800,000 fatal cancers worldwide (the great variance in the numbers depends on what statistical assumptions you make). Presumably you could take such a model and apply it to the southwest to determine its "share" but the small population densities make it very difficult to come up with any "real" numbers attributable to fallout alone, with the exception of Utah, which got comparatively high levels of radioactivity (esp. towns directly downwind, like St. George). (There are likely far more fatal cancers caused by the mining of uranium and associated radon exposure of miners in New Mexico than from atmospheric nuclear testing).