Actually if all you are concerned with is the smoothness of the sample path, the smoothness of a Gaussian process is completely characterized by its covariance function. The following result provides an insight into this issue.
In this aspect we can discuss smoothness with probability one, and the sample path smoothness in this sense is relatively clear in stationary case.
Theorem 3.4.1[Adler] Let $X(\boldsymbol{t}),\boldsymbol{t}\in\mathbb{R}^N$ be a real-valued zero mean Gaussian random field (in your case take $N=1$ as a special case of this result) with a continuous covariance function. Then if for some $0<C<\infty$ and some $\epsilon>0$ $$\boldsymbol{E}|X(\boldsymbol{t}-X(\boldsymbol{s}))|\leq\frac{C}{|\log\|\boldsymbol{t-s}\||^{1+\epsilon}}$$ for all $\boldsymbol{t},\boldsymbol{s}\in I_0\subset\mathbb{R}^N$. Then $X$ has a continuous sample path over $I_0$ with probability one.
In this spirit of "being continuous with probability one", you can actually control the smoothness by choosing different covariance kernels. For example the Matérn covariance function will allow you to have a control over the sample path smoothness by varying its degree of freedom $\nu$; i.e. if you choose a stationary Matérn covariance realization of GP, then the sample path will be in $\mathcal{C}^{[\nu]-1}$ with probability one.
Reference
[Adler] Adler, Robert J. The geometry of random fields. Society for Industrial and Applied Mathematics, 2010.
[Rasmussen] http://ml.dcs.shef.ac.uk/gpip/slides/rasmussen.pdf
