Let us write the Frobenius-Schur indicator of the representation $V$ with highest weight $\lambda$ as $\mathrm{FS}(\lambda)$. Let $R$ denote the root system, and let $\Pi=\{\alpha_1,\dots,\alpha_l\}$ be a basis. Let $\Pi^\vee=\{\alpha_1^\vee,\dots,\alpha_l^\vee\}$ denote the corresponding basis of the dual root system $R^\vee$. Write $\lambda_i=\langle \lambda,\alpha_i^\vee\rangle$.
Exercise 4.3.12 on page 196 of the book by Onishchik and Vinberg "Lie Groups and Algebraic Groups", Springer-Verlag, 1990, says that $$\mathrm{FS}(\lambda)=(-1)^{\sigma(\lambda)},$$ where $$\sigma(\lambda)=\sum_{i=1}^n r_i\lambda_i\tag{*} $$ and $r_i$ are the coordinates of of the sum of positive coroots $2\rho^\vee$ in the basis $\Pi^\vee$ of $R^\vee$. No proof is given.
Another exercise, Exercise 4.3.13, gives an algorithm of computing $\sigma(\lambda)$. Warning: this algorithm contains mistakes.
The coordinates $r_i$ of $2\rho^\vee$ in the basis $\Pi^\vee$ can be found in the tables of Bourbaki "Groupes et algèbres de Lie, Ch. 4,5,6", formula (VII) in each table. Formulas for $r_i\ \mathrm{mod}\ 2$ are given in the last column of Table 3 on page 298 of the book by Onishchik and Vinberg and seem to agree with the formulas of Bourbaki.
For the reader's convenience I give below formulas for $\sigma(\lambda)$ following Onishchik and Vinberg, with the numbering of roots as in Bourbaki. \begin{align*} A_l,\qquad &\sigma(\lambda)=\lambda_1+\lambda_3+\dots+\lambda_l\quad \text{for } l=2q+1,\text{ otherwise } 0;\\ B_l,\qquad & \sigma(\lambda)=\lambda_l\quad\text{for } l=4q+1,\ l=4q+2,\text{ otherwise } 0;\\ C_l\qquad &\sigma(\lambda)=\lambda_1+\lambda_3+\dots\quad\text{ for any } l;\\ D_l\qquad &\sigma(\lambda)=\lambda_{l-1}+\lambda_l\quad\text{for }l=4q+2,\ l=4q+3,\text{ otherwise } 0;\\ E_7\qquad &\sigma(\lambda)=\lambda_2+\lambda_5+\lambda_7\,. \end{align*} For all the other Dynkin diagrams $\sigma(\lambda)=0$ for all $\lambda$ corresponding to self-dual representations.
