It is very important to understand, that there are two cases:
free proton, this is called an inverse beta decay, and it needs an incident anti neutrino of sufficient energy
proton inside a nucleus, in this case the energy is a little bit more (neutrino) then what you would expect based on the mass of the proton and the neutron and the positron
Now the reason the beta plus decay (inside a nucleus) happens, is because of what you are saying, stability. The nucleus becomes more stable by having relatively more neutrons (then before the conversion). The nucleus transforms towards more stability. The reason why it is becoming more stable has to do with the ratio of protons to neutrons. The nucleus is more stable when protons and neutrons number is closer to equal. Why? This has to do with the strong force and the isospin.
The daughter nucleus has a greater binding energy (and therefore a lower total energy) then the mother nucleus. Lower total energy is what every composite particle in the universe (disregarding external effects) is moving towards.
So there are two main reasons the nucleus needs to go to more stability:
isospin, this is very important to understand. In history, the neutron and the proton were considered the same particle, just different isospin. Quarks were discovered, and we know they do have a different internal structure, but the isospin quantum number remains in the SM. Now what is isospin and why do we call it that? Originally, the Pauli exclusion principle was thought (and still does, just is now not so relevant here) to say that two fermions (and neutrons and protons happen to be fermions) cannot occupy the same quantum state (they can't have all quantum numbers the same). So neutrons and protons had to be distinguished. There you have isospin. Now what does it mean after the discovery of quarks and the strong force and the residual strong force? A proton and another proton (or a neutron and another neutron) cannot come very close because:
the Pauli exclusion principle remains
the strong (and the residual strong) force was discovered, and was observed to become repulsive (the residual strong force) at very short distances
Now, why does it matter? Because if they are (neutrons and protons) at optimal distance, the strong force will keep them together with an extra power. What is this extra power? This is the residual strong force's phenomenon, and it causes one neutron and one proton to be able to become more stable (have a stronger bond) then two neutrons or two protons.
This means, that if you have one proton and one neutron, they will align (complement) their isospin, and therefore will become more stable.
If you have two neutrons (or just two protons), these will not be as stable, as if you have one neutron and one proton.
So if you have a nucleus, that has more protons then neutrons (proton-rich), then it will be stable, but it will move towards more stability by equalizing the number of protons and neutrons.
- greater binding energy and lower total energy. A nucleus that has equal number (or closer to equal) of protons and neutrons will have a stronger residual force binding energy (nuclear force) because of isospin and because neutrons and protons are able to come closer then protons and protons (or neutrons and neutrons), and the strong force is very distance dependent, so the smaller nucleus (even if this size difference is very little) will cause a stronger binding energy
Now you are asking what triggers the conversion. This is because you have the misconception that the proton inside the nucleus is stable. It is not. You are looking at this the wrong way. You would think of the proton as a separate entity. It is not. It is part of the nucleus. The nucleus is not stable, because it is proton rich, and it has to transform into a less proton rich nucleus to become more stable. It is the nucleus itself that decays (we do not use the word decay here usually, but transformation). The way the nucleus becomes more stable is by converting the proton into a neutron.
You think of the nucleus as boolean. Stable or not stable. In QM, it is all about probabilities. The proton rich nucleus has its own quantum characteristics, wavefunction, and those give the proton rich nucleus a probability to transform into a more stable energy level, thus become less proton rich. Equilibrium is what the universe wants, and the nucleus is playing along the rules too, wanting to move as close to equilibrium as it can in terms of the number of neutrons and protons.