How does one choose whether or not to draw vertical lines connecting the steps of a step function?
Let's take the cdf of some discrete random variable as an example.
My intuition tells me to graph it this way:
... but many sources have the graphs drawn this way:
Is this a case of six of one, half a dozen of the other? Or is there some important difference?
Your graph is correct. The vertical segments you see in the second graph may be a result of a graphing program that is set in continuous mode so that it connects the dots even when they should not be connected, as in a step function or at a vertical asymptote.
A function $f$ associates a unique value $f(x)$ to each $x$ in its domain. When we graph a function of $x$, we let $y = f(x)$, so there is a unique value of $y$ corresponding to each $x$-value. Consequently, a vertical line can intersect the graph of a function at most once (the Vertical Line Test). Thus, a graph that contains a vertical segment is not a function of $x$.