Let $A$ be a $C^*$-algebra. Let $\{(H_{\lambda}, \varphi_{\lambda})\}_{\lambda \in \Lambda}$ be a family of non-degenerate representations of $A$.
Let $\oplus_{\lambda \in \Lambda} H_{\lambda}$ denote the algebraic direct sum, i.e. subspace of the direct product of the $H_{\lambda}$'s where at most finitely many coordinates are non-zero. This is clearly an inner product space with respect to the canonical inner product:
$\langle (x_{\lambda}), (y_{\lambda}) \rangle := \sum_{\lambda} \langle x_{\lambda}, y_{\lambda} \rangle$.
Define $\varphi:A \to B(\oplus_{\lambda \in \Lambda} H_{\lambda})$ by $\varphi(a)((x_{\lambda})) = (\varphi_{\lambda}(a)(x_{\lambda}))$. It can be checked that $\varphi$ is a well-defined algebra homomorphism. Since each $\varphi_{\lambda}$ is a * - homomorphism between $C^*$-algebras, they are all norm-decreasing.
Let $H$ denote the Hilbert space completion of $\oplus_{\lambda \in \Lambda} H_{\lambda}$. Then we can extend $\varphi$ to the map $\tilde{\varphi}:A \to B(H)$. It can be checked that $\tilde{\varphi}$ is a * - homomorphism and hence $(H, \tilde{\varphi})$ is a representation of $A$.
So, my question is: Is $(H, \tilde{\varphi})$ also a non-degenerate representation of $A$?
Yes. One rather direct way of seeing this is to consider an approximate unit $\{a_j\}$ in $A$. As the net $(\varphi(a_j))$ is a monotone increasing net of selfadjoint operators, it has a strong limit. The non-degeneracy makes it easy to check that $\varphi(a_j)\to 1$. And this implies that $\varphi$ is non-degenerate.