Much more is true: we have $$\mathbb{E} [\exp(\epsilon \|B\|_{0,\alpha}^2)] < \infty$$ for some $\epsilon > 0$. Here $\|\omega\|_{0,\alpha} = \sup_{0 \le s < t \le T} \frac{|\omega(t)-\omega(s)|}{|t-s|^\alpha}$ is the Hölder norm you want.

This is basically Fernique's theorem but we need to be a little more careful because the Hölder space $C^{0,\alpha}([0,T])$ is non-separable. Instead, fix some $1/2 < \beta < \alpha$ and let $X$ be the closure of $C^{0,\beta}([0,T])$ in the $C^{0,\alpha}$ norm. Then $X$ is a separable Banach space (this follows for instance from the compact embedding).

We have $X \subset C([0,1])$ and $X$ has full Wiener measure, so we can view Wiener measure as a Borel probability measure on $X$. One can verify that it is in fact a Gaussian measure. (The issue is that $X$ has a larger dual than $C([0,1])$, and we have to check that these extra functionals still have a Gaussian distribution - but they're weak-* limits of functionals from $C([0,1])^*$.) So now Fernique's theorem applies on $X$ and gives the above result.

Some details can be found in

Baldi, P.; Ben Arous, G.; Kerkyacharian, G., Large deviations and the Strassen theorem in Hölder norm, Stochastic Processes Appl. 42, No. 1, 171-180 (1992). ZBL0757.60014.

in which they prove an even stronger statement, a large deviations principle.

Much more is true: we have $$\mathbb{E} [\exp(\epsilon \|B\|_{0,\alpha}^2)] < \infty$$ for some $\epsilon > 0$. Here $\|\omega\|_{0,\alpha} = \sup_{0 \le s < t \le T} \frac{|\omega(t)-\omega(s)|}{|t-s|^\alpha}$ is the Hölder norm you want.

This is basically Fernique's theorem but we need to be a little more careful because the Hölder space $C^{0,\alpha}([0,T])$ is non-separable. Instead, fix some $1/2 > \beta > \alpha$ and let $X$ be the closure of $C^{0,\beta}([0,T])$ in the $C^{0,\alpha}$ norm. Then $X$ is a separable Banach space (this follows for instance from the compact embedding).

We have $X \subset C([0,1])$ and $X$ has full Wiener measure, so we can view Wiener measure as a Borel probability measure on $X$. One can verify that it is in fact a Gaussian measure. (The issue is that $X$ has a larger dual than $C([0,1])$, and we have to check that these extra functionals still have a Gaussian distribution - but they're weak-* limits of functionals from $C([0,1])^*$.) So now Fernique's theorem applies on $X$ and gives the above result.

Some details can be found in

Baldi, P.; Ben Arous, G.; Kerkyacharian, G., Large deviations and the Strassen theorem in Hölder norm, Stochastic Processes Appl. 42, No. 1, 171-180 (1992). ZBL0757.60014.

in which they prove an even stronger statement, a large deviations principle.