I see no reason to expect that higher voter turnout implies greater political stability. In fact, my intuition is exactly the opposite: assuming a genuine freedom to vote, low turnout is a marker of stability, since it signals that voters don't much care who wins, which suggests that not much of importance depends on the outcome.

This is a really good point, so I'm withdrawing my statement about stability.

To pick another standard meme, what about popular involvement in the political process as a way of promoting just policies over unjust? That is, by unjust policies I mean policies that provide insufficient benefits to people who have little power. This is a separate question from stability, as a stable government can still have extremely unjust policies (or vice versa, though I can't think of examples as easily).