Let $(W_t)_{t\ge 0}$ be a standard Brownian motion starting at zero. Let $f: [0,1]\to\mathbb R$ be a function that is righ-continuous with left limits. Set $$A:=\left\{\omega\in\Omega: \inf_{0\le t\le 1}\big(W_t(\omega)-f(t)\big)=0\right\}.$$ Is $A$ a negligible set? When $f$ is non-decreasing, it is shown by GJC20 in the comments below.

PS : My idea is as follows : Denote by $\tau:=\inf\{t\ge 0: W_t \le f(t)\}$ the hitting time. Then it holds that $$A= \Big(A\cap \{\tau<t\}\Big) \bigcup \Big(A\cap \{\tau=t\}\Big) = :A_1\cup A_2.$$ Clearly $A_2\subset \{W_1=f(1)\}$ is negligible. So it remains to prove so it is with $A_1$. As $$A_1\subset \{\tau<t \mbox{ and } W_{\tau}=f(\tau) \mbox{ and } \exists \epsilon>0 \mbox{ s.t. } W_s \ge f(s) \mbox { for } s\in [\tau+\epsilon, \epsilon] \}=:B,$$ so I wish to show $B$ is negligible. My feeling is to use the fact $f\approx f(\tau)$ uniformly on $[\tau,\tau+\epsilon]$ and the trajectory property of $W$, but I do not know how to make this rigorous.