It is not sequential.  Here is an explicit counterexample.

Notation: $\|\mu\|$ is the total variation norm of $\mu$, and $\bar{B}_r \subset M([0,1])$ is the closed norm ball of radius $r$ centered at $0$.  $\delta_x$ is the Dirac measure which places a unit point mass at $x \in [0,1]$.

For each positive integer $n > 0$, let $P_n \subset [0,1]$ be any finite set of size at least $(2n)^{2n}$, and let
$$E_n = \{ n(\delta_{x} - \delta_y) : x,y \in P_n, \, x \ne y\} \subset M([0,1]).$$

Notice that $E_n$ is finite, and that $\|\mu\| = 2n$ for every $\mu \in E_n$.

Set $E = \bigcup_n E_n$.  I claim that 0 is a weak-* limit point of $E$.  Let $f_1, \dots, f_m \in C([0,1])$ and let $\epsilon > 0$.  I will produce $\mu \in E$ with $\left|\int f_k\,d\mu\right| < \epsilon$ for every $k$.  Without loss of generality, assume $\|f_k\|_\infty \le 1$ for all $k$ (otherwise divide $\epsilon$ by $\max_k \|f_k\|_\infty$).  Define $F : [0,1] \to [-1,1]^m$ by $F = (f_1, \dots, f_m)$.  Choose $n > \max(m, 1/\epsilon)$.  

We can cover $[-1,1]^m$ with $(2n^2)^{m}$ cubes of side length $1/n^2$.  Since $$|P_n| = (2n)^{2n} > (2n)^{2m} = (4n^2)^m > (2n^2)^m$$ by the pigeonhole principle there must exist two distinct $x,y \in P_n$ with $F(x), F(y)$ in the same cube.  This means that $|f_k(x) - f_k(y)| \le 1/n^2$ for every $k$.  So if we take $\mu = n(\delta_x - \delta_y) \in E_n$, we have $\left|\int f_k\,d\mu\right| \le 1/n < \epsilon$ for every $k$.  This proves the claim that 0 is a weak-* limit point of $E$.

Now for any $n$, we know that $\bar{B}_n$ is weak-* closed and disjoint from the finite set $E_n$.  [The weak-* topology is completely regular](http://math.stackexchange.com/questions/352661/is-the-weak-star-topology-on-the-dual-of-a-banach-space-completely-regular), so there exists a weak-* continuous function $G_n : M([0,1]) \to [0,1]$ with $G_n = 0$ on $\bar{B}_n$ and $G_n = 1$ on $E_n$.  Set $G = \sum_{n=1}^\infty G_n$.  Note that on any ball $\bar{B}_n$, only finitely many terms of the sum are nonzero, so the sum makes sense, and the restriction of $G$ to any ball is weak-* continuous.  If $\mu_k$ is a weak-* convergent sequence with limit $\mu$, then by the uniform boundedness principle all the $\mu_k$ and $\mu$ lie in some ball $B$.  The restriction of $G$ to $B$ is weak-* continuous, so $G(\mu_k) \to G(\mu)$.  Thus we have shown $G$ is weak-* sequentially continuous.

On the other hand, $G$ is not weak-* continuous, since $G(0) = 0$ but $G \ge 1$ on $E$.


(The same uniform boundedness argument shows that $E$ is weak-* sequentially closed, but not weak-* closed since it does not contain 0.)

More generally, we can replace $M([0,1])$ by the dual of any separable Banach space.  See the paper:

> Humphrey, A. James and Simpson, Stephen G.  Separable Banach space theory needs strong set existence axioms.  *Trans. Amer. Math. Soc.* **348** (10), 4231-4255, 1996.  [Open access full text](http://www.ams.org/journals/tran/1996-348-10/S0002-9947-96-01725-4/S0002-9947-96-01725-4.pdf)

Theorem 2.5 of that paper shows that for any infinite-dimensional separable Banach space $X$, there is a countable subset $Z \subset X^*$ which is weak-* sequentially closed but weak-* dense.  In particular, $Z$ is not weak-* closed, so the weak-* topology on $X^*$ is not sequential in the usual sense of the word.  

To show it is not sequential in your sense: As an intermediate step of their construction, they get a sequence which has $0$ as a weak-* limit point but intersects every ball in only finitely many points; we could proceed as we did above to construct a function $G : X^* \to \mathbb{R}$ which is weak-* sequentially continuous but not weak-* continuous.  (Their set $Z$ is formed by translating such a sequence by a countable weak-* dense set.)