Call the bounded linear functionals $\Lambda_r$.
Apply the Banach-Steinhaus theorem. We either have $E \subset X$ (a dense $G_\delta$) for which
$ \sup_{r \in \mathbb{Q} \cap [0, 1]} \left|f(r)\right| = \infty $
for all $f \in E$. Or there exists $M < \infty$ such that
$ \|\Lambda_r\| \le M $
for all $r \in \mathbb{Q} \cap [0, 1]$. Since $X$ is a subset of $C([0, 1])$, all of its functions are bounded and the first case is impossible. Hence
$ \left|f(r)\right| \le M \|f\| $
for all $r \in \mathbb{Q} \cap [0, 1]$ and for all $f \in X$. Since the set $\mathbb{Q} \cap [0, 1]$ is dense in $[0, 1]$ and all functions in $X$ are continuous, the desired result follows.