At first blush, I wanted to give this suggestion a +10. But then I realized that was my own bias talking, rather than a decision stemming from an objective assessment of the situation.
I think Apple's decision to hide the scrollbars by default was a terrible, horrible decision from a UX perspective. It's perfectly reasonable on screens with really limited real estate like an iOS device, but it makes absolutely no sense on the Mac where screens are getting nothing but larger. I haven't upgraded my notebook to Lion yet (still using Snow Leopard because I need Rosetta), but when I do, I'll certainly be disabling that "feature" once and for all.
But therein lies the rub… This is a preference that can be configured in the operating system for a reason: because some people like to have their scrollbars hidden (for some inexplicable reason) and other people don't. Sure, you and I and other rational people fall squarely into the latter category, but that doesn't give us the right to impose our preferences on others. If a user has configured their operating system not to display scrollbars, then the Stack Exchange CSS should respect those preferences, rather than try to override them.
As I see it, the worst case scenario is when you have three different options. Instead of just showing or not showing scrollbars, you can now have scrollbars shown on Stack Exchange sites in the code blocks and hidden everywhere else? Objectively speaking, that's even more confusing and broken.
So my verdict is a no. This suggestion breaks the cardinal rule: always respect the user's preferences.
However, an alternate solution comes to mind. Rather than forcing the scrollbar to be persistently shown, we could solve the problem of ambiguity in a different way.
The issue, as you point out, is that it's not always clear if there is more code to be displayed than will fit inside of the code block in the case where scrollbars are not being shown. We could fix this by adding a different sort of visual cue to the code well.
For example, if there's additional code that won't fit, a bevel or shadow could be added to the bottom (or top) of the code well, suggesting that there's more there to be seen if you scroll down. This gives us the visual affordance we want, without overriding the user's preferences.
But I have no idea if this is even possible from a technical perspective.