A new class of infinite dimensional representations of the Yangians corresponding to a complex semisimple algebra and its Borel subalgebra is constructed. It is based on the generalization of the Drinfeld realization of the Yangians in terms of quantum minors to the case of an arbitrary semisimple Lie algebra. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of monopoles defined as the components of the space of based maps of one-dimensional complex projective space into the generalized flag manifold. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles.