Looks like plots became of low quality in Mathematica 9?

How to make Mathematica to plot continuously? Logically current plot style is solid.
Interestingly, your plot rendered fine as per my comments, and more interestingly, as soon as I added an Export after the fact, the plot looked like yours. This persisted even after removing the export and re-plotting: I had to quit the kernel, restart, and re-plot. Might be a weird bug, I will be investigating further.
In any case, DiscretePlot is really better suited to the CDF of a discrete distribution:
DiscretePlot[CDF[BinomialDistribution[1001, 1/2], x], {x, 0, 1001}, PlotRange -> {0, 1}, Filling -> None, Joined -> True] 
Exclusions -> Noneso it will be a continuous stairs like plot $\endgroup$