Fix $p$. Leopoldt's conjecture is that the $p$-adic regulator does not vanish. Since that is an open condition in the $p$-adic numbers, the set of fields on which the conjecture is true ought to be open. That sounds like a pretty good avatar of your question, although it doesn’t give density like the Zariski topology. Also, this is only a conjecture.

What does that even mean? Define the $p$-adic topology on the set of number fields by way of test curves, which are parameterized by finite extensions of $\mathbb Q(t)$. Each unramified rational value of $t$ gives an extension of $\mathbb Q$. So each such test curve defines a set of number fields and identifies it with a cofinite subset of $\mathbb Q$. We can transfer the $p$-adic topology on $\mathbb Q$ to the set of number fields. A subset of fields is $p$-adically open if its preimage in each test curve is open. A function of number fields is $p$-adically continuous if its pullback to each test curve is.

Is the $p$-adic regulator a continuous function of number fields? I doubt it. Consider the test curve $x^2=t$. For negative $t$ there are no units, so the regulator is 1. For positive $t$, it isn’t. So probably the test curves are too big and we should make the finite partition according to Archimedean behavior. But even after restricting to real quadratics, it is still unclear because the fundamental unit is hard to control because of the class group.

But there are some test curves for which we can control specific units to make a continuous proxy for the regulator, and thus prove that the set of points satisfying Leopoldt’s conjecture is open (though maybe empty). Specifically, the Ankeny-Brauer-Chowla family is the extension of $\mathbb Q(a_1,\ldots,a_n)$ defined by $\prod (x-a_i)=1$. Each specialization is a degree $n$ totally real field with $n-1$ independent units easily described in terms of the generator. The behavior of the extension of $\mathbb Q_p$ is locally constant, so we can locally think of the $n-1$ units as varying continuously through a single $p$-adic algebra. Thus the logarithms of the units vary continuously and so do their minors. This isn’t quite the regulator, since these units don’t necessarily generate all units, but a maximal minor avoiding zero is equivalent to Leopoldt’s conjecture, so the conjecture is open on this test variety.

I got this all from here, which also discusses a generalization (indeed, a deformation) of ABC fields to fields with imaginary places.