The following result of Howe [https://zbmath.org/0844.20027] answers this completely:

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra.  Then a non-trivial irreducible $\mathfrak{g}$-module $V(\lambda)$ has one-dimensional weight spaces if and only if 

 1. $\lambda$ is minuscule, 
 2. $\lambda$ is quasi-minuscule and $\mathfrak{g}$ has only one short simple root, 
 3. $\mathfrak{g}=C_{3}=\mathfrak{sp}_{6}$ and $\lambda=\omega_{3}$, or 
 4. $\mathfrak{g}=A_{l}=\mathfrak{sl}_{l+1}$ and $\lambda=m\omega_{1}$ or $\lambda=m\omega_{l}$ for some $m\in \mathbb{N}$.

I was able to find this via a paper of Stembridge [https://zbmath.org/1060.17001].

As I needed exactly this property in a paper of my own, you will find definitions of minuscule and quasi-minuscule as well as references from which I drew a table of these on pages 20-21 of it (arXiv:0409359, https://arxiv.org/abs/math/0409359).  They are as follows, taken from [https://zbmath.org/0957.17006] by Plotkin--Semenov--Vavilov:

Non-zero minuscule weights:

$\begin{array}{ll}  	
			A_{l} & \omega_{i},\ 1\leq i \leq l \\ 
			B_{l} & \omega_{l} \\ 
			C_{l} & \omega_{1} \\ 
			D_{l} & \omega_{1},\ \omega_{l-1},\ \omega_{l} \\ 
			E_{6} & \omega_{1},\ \omega_{6} \\ 
			E_{7} & \omega_{7} 
\end{array} $

Quasi-minuscule weights:

$ \begin{array}{ll} 
\begin{array}[t]{ll}  	A_{l} & [1,0,0,\ldots,0,1]\ (\mbox{adjoint}) \\ 
			B_{l} & \omega_{1} \\ 
			C_{l} & \omega_{2} \\ 
			D_{l} & \omega_{2} \\ 
			E_{6} & \omega_{2} \\
E_{7} & \omega_{1} \\ 
			E_{8} & \omega_{8} \\ 
			F_{4} & \omega_{4} \\ 
			G_{2} & \omega_{1} 
\end{array}
\end{array} $