What are the minimal conditions on a topological space for its closure to equal the sequential closure? My question is: what are the minimal conditions on a topological space for it have the following property? 
$$x\in \bar{A}\iff \exists (x_n)\subset A | x_n \to x$$

In this paper there is the answer (section 2, on Fréchet spaces, also known as Fréchet-Urysohn spaces): Your property defines the notion of a Fréchet space and he shows that these spaces are the pseudo-open images of metric spaces. He also defines the weaker concept of a sequential space and in the follow up paper he shows that a sequential space is Fréchet iff all of its subspaces are sequential (hereditarily sequential).
A sequential space has the cleaner characterization: a space $X$ is sequential iff there is a metric space $M$ and a surjective quotient map $f: M \rightarrow X$ ($X$ is a quotient image of a metric space).
As said, Fréchet spaces can be similarly characterized, using not quotient maps but pseudo-open maps: $f: X \rightarrow Y$ is pseudo-open iff for every $y \in Y$ and every open neighborhood $U$ of $f^{-1}[\{y\}]$ we have that $y \in \operatorname{int}(f[U])$. Every open or closed surjective map is pseudo-open and all pseudo-open maps are quotient.  
