Fundamentally, the requirement for the shaft according to ASME Y14.5 is that it shouldn't exceeds the maximum dimension (MMC) envelope and in no individual location should it be less than the the minimum (LMC). As someone above correctly identified, the proper way would to be with a ring gage (really a long tube to be 100% accurate) to establish that it doesn't exceed the maximum dimension and it should be measured along the length and around the shaft w/ a 2 point measurement system (micrometer, etc) to make sure that at no point it is less than the minimum requirement.
That being said, often in reality one would take some sort of approximation by only taking one or several measurements. What it comes down to is how much variation you expect/fear in form of the part. If you expect the diameter to be uniform along the length, perfectly round, and perfectly straight, than a single diameter measurement would be sufficient. The more types of form error you're trying to inspect for, the more you need to broaden how many measurements (and what locations) you take and how you make them. Perhaps on initial production parts, you could do the full gammut of inspections, and once you have better understanding on the form variation and the nature of the manufacturing process, you can simplify the measurement.
As an aside, someone be above mentioned averaging of values (and using CMMs); id caution against doing that unless the print calls explicitly for an average diameter. We've been horribly bitten by this method in the past as it overlooks the true requirements. In our case, we were using a CMM and had (unbeknownst to us) oval shaped holes that in one direction were larger than the maximum hole size while in the opposite direction were smaller than the minimum size. The CMM took the points and curve fit a best fit circle to the group coming up with an average circle that fell within requirements but was really nonconforming. We ended up showing this by being unable to fit in MMC pin gages into the hole and also making multiple measurements w/ a 2 point bore gage. We learned the hard way to the importance of not using the CMM's average. I've pretty much tried to eliminate averaging, it rarely represents what we really care about in the parts - it is much more important to report the maximum and minimum. The only time i still even consider it is when the measurement uncertainty is particularly large (usually a chemical/analytical test) and I'm trying to minimize the gage error.
David