I'll summarize what I've learned.
Denote a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_{1},t_{2})=k(|t_{1}-t_{2}|)$ defined on the interval $[0,T]$ and also $m=\inf_{t\in[0,T]}g(t)$.
I could not find any result regarding the pdf (or cdf) of $m$, not even its mean. Also $P(m>c)$ yielded no results except for the case $c=0$, where from symmetry $P(m>0)=\frac{1}{2}p(0,T)$ with $p(n,T)$ is the probability of $g(t)$ having exactly $n$ zeros in $[0,T]$. This latter quantity is discussed extensively in the literature on “zero crossing intervals” and can be calculated numerically, as we shall now explain.
As already mensioned, the mean number of zeros in time unit is given by Rice' Formula $\beta=\sqrt{-k''(0)}/\pi$ (when $k$ is normalized, $k(0)=1$). Also denote $P_{n}(\tau)$ the probability density for of the event that the $n+1$th zero crossing occuring at interval $\tau$ after the first zero crossing.
McFadden's paper from 1958, The axis-crossing intervals of random functions--II
show that $U(\tau)=\sum_{0}^{\infty}P_{n}(\tau)$, the probability of some zero crossing at interval $\tau$ after a zero crossing have a (complicated) closed-form expression $U(\tau)=\frac{2}{\pi}(\frac{1}{H(\tau)}-\tan^{-1}H(\tau))R(\tau)$ (eq. 20-21 there) with $R(\tau)=\frac{1}{2\beta\pi}\frac{k(\tau)k'(\tau)^{2}+(1-k(\tau)^{2})k''(\tau)}{(1-k(\tau)^{2})^{3/2}}$ and $H(\tau)=\frac{k(\tau)k'(\tau)^{2}+(1-k(\tau)^{2})k''(\tau)}{\sqrt{[k'(\tau)^{2}+(1-k(\tau)^{2})k''(0)]^{2}-[k(\tau)k'(\tau)^{2}+(1-k(\tau)^{2})k''(\tau)]^{2}}}$. Now $P_{0}(\tau)$ can be calculating numerically from the convolutional equation $P_{0}(\tau)=\frac{1}{2}[U(\tau)+R(\tau)]-\frac{1}{2}[U(\tau)-R(\tau)]*P_{0}(\tau)$ (eq. 47 there). Then $p(0,T)$ can be calculated numerically by integrating $p(0,T)=1+\beta[\int_{0}^{T}d\tau\int_{0}^{\tau}dlP_{0}(l)-\tau]$.
