Magnonics has been receiving significant attention in magnetism and spintronics because of its premise for devices using spin current carried by magnons, the quanta of spin-wave excitations of macroscopically ordered magnetic media. Although magnonics has a clear energywise advantage over conventional electronics due to the absence of Joule heating, inherent magnon-magnon interactions give rise to a finite lifetime of magnons, which has been hampering the efficient realization of magnonic devices. To promote magnonics, it is imperative to identify the delocalized magnon modes that are minimally affected by magnon-magnon interactions and thus possess a long lifetime and use them to achieve efficient magnon transport. Here, we suggest that quasicrystals may offer the solution to this problem via critical magnon modes that are neither extended nor localized. We find that a critical magnon exhibits fractal characteristics that are absent in conventional magnon modes in regular solids such as a unique power-law scaling and a self-similar distribution of distances showing perfect magnon transmission. Moreover, critical magnons have longer lifetimes compared to the extended ones in a periodic system, by suppressing the magnon-magnon interaction decay rate. Such an enhancement of the magnon stability originates from the presence of the quasiperiodicity and intermediate localization behavior of critical magnons. Thus, we offer the utility of quasicrystals and their critical spin-wave functions in magnonics as unique fractal transport characteristics and enhanced stability.