  We study Hausdorff-Young type inequalities for vector-valued Dirichlet series
which allow to compare the norm of a Dirichlet series in the Hardy space
$\mathcal{H}_{p} (X)$ with the $q$-norm of its coefficients. In order to obtain
inequalities completely analogous to the scalar case, a Banach space must
satisfy the restrictive notion of Fourier type/cotype. We show that variants of
these inequalities hold for the much broader range of spaces enjoying
type/cotype. We also consider Hausdorff-Young type inequalities for functions
defined on the infinite torus $\mathbb{T}^{\infty}$ or the boolean cube
$\{-1,1\}^{\infty}$.
