The Borel $\sigma$-algebra on the space $C(K,E)$ of continuous functions from a compact metrizable space $K$ to a separable metric space with the induced uniform metric is generated by the evaluation maps of the form $f\mapsto f(x)$. See, for example, Lemma 4.53 in Aliprantis & Border (2006).
Therefore, your random function is measurable if for every $x\in K$, the value at $x$ is a random variable.