You can use a function called downloadHandler for this purpose. Something like this:
output$downloadPlot <- downloadHandler(
    filename = function() { paste(input$dataset, '.png', sep='') },
    content = function(file) {
        device <- function(..., width, height) grDevices::png(..., width = width, height = height, res = 300, units = "in")
        ggsave(file, plot = plotInput(), device = device)
    }
)