I don't have the text of Drinfeld's address near to hand, but the standard way I know to do this is to take the subalgebra of the (finite) dual generated by the matrix coefficients of the irreducible finite-dimensional representations of the quantized enveloping algebra. I mostly work with the $q$-deformed versions though, so perhaps there are some subtle differences in the formal power series version.

For the infinite series of simple Lie algebras over $\mathbb{C}$, you can also construct the quantized function algebras a bit more abstractly. The idea is that you start with a matrix coalgebra, i.e the span of some elements $u^i_j$ with coproduct $\Delta(u^i_j) = \sum_k u^i_k \otimes u^k_j$ and counit $\epsilon(u^i_j) = \delta_{ij}$. You then take the tensor algebra over this matrix coalgebra, which is naturally a bialgebra, and quotient by the quadratic relations
$$ R^{ji}_{kl} u^k_m u^l_n = u^i_k u^j_l R^{lk}_{mn},$$
where $R$ is the $R$-matrix for the vector representation of the quantized enveloping algebra that you started with.

The result of this is called an FRT-bialgebra; you have to take a further quotient by a quantum determinant to get a Hopf algebra. Then you can show that the Hopf algebra you've constructed has a nondegenerate Hopf pairing with the quantized enveloping algebra.

Now that I think about it, though, it doesn't seem like these two constructions will give the same result in all cases - e.g. for $\mathfrak{so}_{2N}$, where not all of the irreps appear as subreps in some tensor power of the vector representation. But I guess in that situation, the first construction would give you the quantized function algebra of the simply connected cover of the group, whereas the second construction would yield the quantized function algebra of $SO(2N)$.

My standard reference for this stuff is Klimyk and Schmudgen's book Quantum Groups and Their Representations. The quantized function algebra business is in Chapter 9.