We study magnetic skyrmions in a two-dimensional quasicrystalline lattice using a classical Heisenberg model with Dzyaloshinskii-Moriya interactions and an external magnetic field. The competition between the skyrmion-skyrmion repulsion and an emergent quasiperiodic pinning landscape gives rise to a sequence of distinct skyrmion lattice configurations as a function of field. The resulting hierarchy of quasiperiodic pinning potentials, characterized by closely spaced quasi-degenerate minima, enables a quasi-continuous suppression of the skyrmion density as the saturation field is approached, in sharp contrast to the strongly first-order collapse of skyrmion crystals on periodic lattices. This provides a direct mechanism for controlling the topological charge and, consequently, the emergent gauge field for itinerant electrons. As a consequence, the Hall conductivity can be strongly modified with small changes in the magnetic field and driven smoothly to zero near saturation. This field-controlled tunability, rooted in the underlying multistability, identifies quasicrystalline magnets as a platform for tunable topological textures, with potential applications in magnetic memory and magnetoelectronic response.