As R W says, the answer is in Plante's paper, but the special case of $1$-dimensional foliations is actually simple, and well-known: let $M$ be the manifold, $*\in M$ be a basepoint, $(\phi^t)$ be the flow (assumed tangential to the boundary of $M$, if any). For every integer $n\ge 0$, let $\mu_n$ be the image in $M$ of the probability measure $dt/(2n+1)$ on the interval $[-n,+n]$ under the map
$t\mapsto\phi^t(*)$.
Since the space of Borelian probability measures on $M$ is compact for the weak topology, the sequence $(\mu_n)$ has a subsequence weakly converging to a Borelian probability measure $\mu$ on $M$.
Claim: $\mu$ is invariant by every $\phi^t$ (which amounts to a transverse invariant measure). Indeed, for a fixed $t$, the sequence of measures $$\phi^t_*\mu_n-\mu_n$$ goes to $0$ in the norm topology, since for any continuous real function $f$ on $M$, one has:
$$\vert(\phi^t_*\mu_n-\mu_n)f\vert=\frac{1}{2n+1}\vert\int_{-n+t}^{n+t}f(\phi^t(s))ds-\int_{-n}^{+n}f(\phi^t(s))ds\vert\le\frac{2t}{2n+1}\Vert f\Vert_\infty$$