Let $S$ be a complex non-singular projective surface and let $C$ be a complex non-singular projective curve. Moreover consider a morphism $\varphi:S\longrightarrow C$ which is flat, proper and with connected fibers.
Is it true that the subset $$U=\{p\in C\,:\, \text{the fiber $S_p$ is a smooth curve}\}\subseteq C$$ is an open subset of $C$?
Yes that is true. See Hartshorne Ch.III Corollary 10.7. In fact, you don't need half of the assumptions above, it is a more general fact. It works if $S$ is nonsingular surface over $\mathbb{C}$ and $C$ is any complex curve and $\varphi$ is a separated morphism with connected fibers.