Will a vertex present in the set of vertices present in each basis cycle be present in all the cycles?
Consider the following graph:



Let $S$ be the set containing the cycles formed by i) the red and black edges, and ii) the blue and black edges. $S$ is a cycle basis in which every cycle contains vertex $x$, yet the cycle formed by the red and blue edges does not contain $x$. 



Edit: it is also not true that the existence of a vertex $x$ that lies in all cycles of a cycle basis implies that $x$ belongs to all cycles of the graph. See the following counterexample:



The $3$ cycles that can be formed by taking an edge $e$ of color $c$ for $c \in \{ \mbox{red, green, blue}\}$ along with the two black edges incident to the endpoints of $e$ form a basis $S$. Vertex $x$ belongs to all the cycles in $S$ but it does not belong to the cycle formed by the red, green, and blue edges.