Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha}: \alpha\in S\}$ such that $\overline{A}=\overline{\bigcup A_{\alpha}}$ (where $\overline{A}$ denotes the closure of $A$). Can anyone find an example of a space which has the above property but is not zerodimensional?

Any space which contains a dense set of isolated points will have the property. It is easy to construct (e.g. as subspaces of $\mathbb{R}^n$) such spaces with arbitrarily large dimension. For a one dimensional explicit construction see B.M.Scott's answer to this question: http://math.stackexchange.com/questions/152390/cardinalityofadenseopenset. 


Yes, there are spaces where this property holds which are not zerodimensional: consider the twopoint space $\lbrace a, b\rbrace$ with topology $\lbrace \lbrace a\rbrace, \lbrace a, b\rbrace, \emptyset \rbrace$. Then this space clearly does not have a basis of clopen sets, so is not zerodimensional, but the only closed sets are $\emptyset$ and $\lbrace a, b\rbrace$ which are both the closures of clopen sets. Note that a strengthening of your property  where we require the clopen $A_\alpha$ to be subsets of $A$  does not hold in this case, and I suspect that stronger version is equivalent to being zerodimensional. [EDIT: as Ramiro points out, this is not the case; see comments] Also, note that there are multiple definitions of "zerodimensional," so you should specify which one you mean. (I am still curious what the motivation for this problem is.) 


It is Easy to see that $X$ has my property if and only if $\beta X$ has my property. So if we consider Dowkers example $Y$ then $Y$ is a zerodimensional where $\beta Y$ is not zerodimensional (see general topology book, Ryszard Engelking). On the other hand $\beta Y$ has my property. 

