The set of Bernoulli measures $B$ is closed, and the closure of $C$ is precisely the set of measures of the form $\int m\,d\mathbb{P}$ where $\mathbb{P}$ is a Borel probability measure on the set of Bernoulli measures.
Let me unpack that statement a little. Let $\mathcal{M}_\sigma$ denote the set of shift-invariant Borel probability measures on $\Sigma_d:=\{1,\ldots,d\}^{\mathbb{N}}$. This we equip with the weak-* topology, which is the smallest topology on $\mathcal{M}_\sigma$ such that for every continuous function $f \colon \Sigma_d \to \mathbb{R}$, the map $m \mapsto \int f\,dm$ is a continuous function from $\mathcal{M}_\sigma$ to $\mathbb{R}$.
This topology (which is metrisable) defines a Borel $\sigma$-algebra on $\mathcal{M}_\sigma$. A linear combination of measures $\sum_{i=1}^n\alpha_im_i \in \mathcal{M}_\sigma$ satisfies
$$\int f\,d\left(\sum_{i=1}^n \alpha_i m_i\right) = \sum_{i=1}^n \alpha_i \int f\,dm_i$$
for every $f \in C(\Sigma_d)$. More generally, if $\mathbb{P}$ is a Borel probability measure on the compact metrisable space $\mathcal{M}_\sigma$, then defining
$$\int f\,d\mu:=\iint f \,dm\,d\mathbb{P}(m)$$
defines a measure $\mu$ on $\Sigma_d$, which we abbreviate to $\mu=\int m\,d\mathbb{P}(m)$.
Now, it is easy to check that the set of Bernoulli measures is closed in the weak-* topology: one uses the fact that a measure is $(p_1,\ldots,p_d)$-Bernoulli if and only if every cylinder set satisfies an equation in the weights,
$$\mu([x_1\cdots x_n])=p_{x_1}\cdots p_{x_n},$$
and $\mu([x_1\cdots x_n])$ is just $\int \chi \,d\mu$ where $\chi$ is the (continuous) indicator function of the (clopen) cylinder set $[x_1\cdots x_n]$. The set $C$ is thus the set of all measures of the form $\int m\,d\mathbb{P}$ where $\mathbb{P}$ is an atomic probability measure supported on the closed set $B$. Since $B$ is closed, the closure of $C$ is the set of all measures of the form $\int m\,d\mathbb{P}$ where $\mathbb{P}$ is an arbitrary Borel probability measure on $B$.
I am not quite sure how to answer the question "How big is the closure of $C$?" but I can at least say that it is an infinite-dimensional sub-simplex of $\mathcal{M}_\sigma$, and its only ergodic elements are Bernoulli measures.
