Every metric space with a countable base is separable I know that every separable metric space has a countable base.

I was wondering if we can get a countable dense subset from every metric space that has a countable base.

Thank you very much!!
You don't even need a metric space for that direction, in holds in general. If your topology has a countable base, just pick one element out of every base set and you will get a countable dense subset.
In general in any topological space  let $A=\{a_i : a_i \in O_i\}$ you  can prove that $A$ is a dense countable subset, such that $B=\{O_i : i \in\mathbb N\}$ is the Base of topological space