At the fundamental level, $\beta^-$-decay is: $$d \to u + e^- + \overline {\nu_e}$$ and this decay occurs spontaneously.
From spontaneousness and mass-energy equivalence, can I say that down quark is heavier(rest mass, current quark mass) than up quark? Or is there other factors involved?
As it happens the mass of the down quark is greater than the mass of the up quark, though you cannot take beta decay as evidence for this.
The proton and neutron are fearsomely complicated beasts. Their mass of around a Gev is a factor of a hundred times greater than the mass of the three valence quarks inside them. The extra 99% of the mass is made up from the interaction energies of the various components present inside the particle such as the gluons and virtual quarks.
This means you cannot simply treat the down quark in isolation from everything else. For example the difference in mass between the neutron and proton, $\approx 1.3$MeV, is about half the difference in mass between the down and up quarks $\approx 2.5$MeV so changes in the interaction energies are also involved in beta decay.
So all you can say is the the total masses and interaction energies in the neutron are greater (by $1.3$MeV) than the proton, and this 1.3Mev difference goes into the rest masses and kinetic energies of the electron and anti-neutrino produced by the beta decay.
The extra energy is a part of the reason why beta decay happens. However it is not the only reason it decays. The reason the down quark in the neutron actually decays has to do with what baryon is the most stable. It turns out the proton is the most stable baryon. So every other baryon must decay into the proton. The extra mass of the down quark is part of the reason. However the way that down quark interacts with gluons, and virtual quark and anti quark pair also has something to do with this.
