If X is a random variable witch for some $\theta > 0$, $\mathbb{E}[e^{\theta X}] < 1$, follow from Jensen's Inequality, that $$\theta \mathbb{E}[X] = \mathbb{E}[\log(e^{\theta X})] \leq \log \left( \mathbb{E}[e^{\theta X}] \right) < 0.$$ So ${\mathbb{E}[X] < 0}$.
The converse is also true?
I need this information to apply the Crámer's Theorem in a problem where $\mathbb{E}[X] < 0$.
Yes, so long as the moment-generating function exists in a neighborhood of $0$. Let $\Phi(\theta) = \mathbb E[e^{\theta X}]$; recall (somewhat unsatisfying reference here) that if $\Phi(\theta)$ exists, then it is continuously differentiable in a neighborhood of the origin. Recall also that $\mathbb E[X] = \Phi'(0)$. You can use the facts that $\Phi(0) = 1$ and $\Phi'(0) < 1$ along with continuous differentiability and the Mean Value Theorem to argue that there is some positive $\theta$ for which $\Phi(\theta) < 1$.