If $X$ is a baire space, then every countable interesction of open dense sets is dense in $X$.
Now, assume that $X$ is some topological space with the propery:
If $A_{1},A_{2},...$ are open dense sets in $X$, then $\bigcap _{n=1} ^{\infty} A_{n} \ne \emptyset$.
I'm looking for space $X$ which respects that property, but is not a Baire space. In other words, i'm looking for a space $X$ such that every countable intersection of open dense sets is non-empty, but not necessary dense.
Let $X$ be the disjoint union of $\Bbb Q$ and an isolated point. Every countable intersection of dense open sets must contain the isolated point, but its singleton is not dense and is the intersection of a countable family of dense open sets.
Equivalently, let $X=\big(\Bbb Q\cap(0,1)\big)\cup\{2\}$ as a subspace of $\Bbb R$.
A slightly less trivial example is $X=\big(\Bbb Q\cap(0,1)\big)\cup[2,3]$ as a subspace of $\Bbb R$: every countable intersection of dense open sets will be dense in $[2,3]$ but need not be dense in $X$.
