For the very limited situation of $k=1$, the VC dimension is $\lfloor \log_2(n+1) + 1 \rfloor$.

The upper bound follows essentially from Sasho's argument: there are $2+2n$ 1-juntas on $n$ variables, and if there were more than $\log_2$ of that many inputs, we could find a function that avoids all 1-juntas.

To see the lower bound, consider the matrix whose columns contain the $2^{\lfloor \log_2(n+1) \rfloor} - 1 \le n$ not-all-zero bitstrings of length $\lfloor \log_2(n+1) \rfloor + 1$ that start with a zero. The inputs to shatter are the rows of this matrix (padded up to length $n$). Every function of these inputs is either constant, or else it or its negation appears as a column, so it's computed by a 1-junta.

This idea can in theory be extended to larger $k$, where you encode every function as some $k$ columns in the matrix. I don't see how best to argue this at the moment, though.