Without directly answering the question about Drinfeld's paper, I'd emphasize the variations in the definition of "quantum group" in the literature and the resulting variation in discussions about centers, representations, etc. The early work by Drinfeld and Jimbo was partly motivated by mathematical physics, so the $q$ in the definition might be a complex number (say related to the Planck constant). But in later work by Lusztig and others the $q$ is at first an indeterminate and can then be specialized, so that $q=1$ recovers something close to the universal enveloping algebra of a complex semisimple Lie algebra whereas $q \neq 1$ a root of unity leads elsewhere. Jantzen's book mostly follows Lusztig's lead in the treatment of quantized universal enveloping algebras, where a version of Kostant's $\mathbb{Z}$-form leads to finite dimensional versions, etc. (Other generalizations involve affine Lie algebras, while very different "quantum groups" arise from function algebras of algebraic groups.)

In all of these settings it is a problem to describe the center of the quantum group and relate it to the (initially finite dimensional) representation theory.
As far as I know, only partial information about centers has been developed so far, while at least for Lusztig's quantum groups the finite dimensional representations have mostly been studied without full knowledge of the center.
That has certainly been true in modular representation theory as well.

The good classical prototype occurs in the older work of Chevalley and Harish-Chandra on the usual universal enveloping algebra of a complex semisimple Lie algebra (say of rank $\ell$). Here the center turns out to be just a polynomial algebra in $\ell$ indeterminates, closely related to the enveloping algebra of a Cartan subalgebra and characterized as a suitable algebra of Weyl group invariants. In turn, the "central characters" are easily described and help to sort out representations (as in the earlier use of isolated Casimir elements)
even in the more complicated infinite dimensional setting of the BGG category.
The notion of "infinitesimal character" even plays a role in the study of Lie group representations, but is only one of many ingredients there. Note that trace functions arising from representations are (as in Drinfeld's construction) a natural way to relate the center to the representation category.

In prime characteristic things get more complicated, as in Lusztig's work on quantum groups at a root of unity: here the finite Weyl group tends to give way to an infinite Coxeter group such as the affine Weyl group (of Langlands dual type) and much is still not understood even though the quantum group representations have been fairly well sorted out. Along the way the role of the center gets diminished, though is still potentially quite interesting.

If the question here is limited just to the Lusztig version of quantized enveloping algebras in characteristic 0 (for an indeterminate $q$), I'm not sure the center has yet been understood well enough from Drinfeld's viewpoint to contribute much to the study of finite dimensional representations. The latter closely resemble the familiar highest weight representations of a semisimple Lie algebra and can be viewed as "quantizations".

By now there is of course a lot of literature to consult, beyond the original papers by Drinfeld and Jimbo. Jantzen gives a good introduction to Lusztig's theory, while Lusztig's many papers (and one book) go farther. But the centers of the various Hopf algebras remain mysterious, to me at least.

ADDED: Probably the most useful book to consult is the 1994 Cambridge treatise *A Guide to Quantum Groups* by Chari and Pressley (corrected paperback reprint in 1995), which also has an extensive bibliography. They follow some of Drinfeld's 1989/1990 paper for their general discussion in 4.2A of "almost cocommutative" Hopf algebras. Chapter 10 deals with Lusztig's formulation of the quantum enveloping algebra, with $q$ an indeterminate. Here the work of Marc Rosso brings out the classical-looking role of the center in the study of finite dimensional representations. See especially Rosso's 1990 *Ann. Scient. Ecole Norm. Sup.* paper (and related Bourbaki seminar talk), both available online at www.numdam.org by doing a quick search for "Rosso".