First, the strong force acts on scales where our classical idea of forces as something that obeys Newton's laws breaks down anyway. The proper description of the strong force is as a quantum field theory. On the level of quarks, this is a theory of gluons, but on scales of the nucleus, only a "residual strong force", the nuclear force remains, which can be thought of as being effectively mediated by pions.
Now, a force mediated by pions is very different from one mediated by photons, for the simple reason that pions are massive. Massive forces do not, in their classical limit, follow a pure inverse square law, but yield the more general Yukawa potential, which goes as $\propto \frac{\mathrm{e}^{-mr}}{r^2}$ where $m$ is the mass of the mediating particle. That is, massive forces fall off far faster than electromagnetism.
So this makes it already difficult to tell what the "strength" of a force exactly is - it depends on the scale you are looking at, as Wikipedia's table for the strengths of the fundamental forces rightly acknowledges. However, in no sense is the strong force "infinitely stronger" than the electromagnetic force - it is simply much stronger than it, sufficient to keep nuclei together against electromagnetic repulsion.
Now, the person who said that it is "infinitely stronger" might have had something different in mind which is not actually related to the strength of the force but to its fundamentally quantum mechanical nature: Confinement, the phenomenon that particles charged under the fundamental (not the residual) strong force cannot freely exist in nature. When you try - electromagnetically or otherwise - to separate two quarks bound by the strong force, then you will never get two free quarks. The force between these two quarks stays constant with increasing distance, it does not obey an inverse square law at all, and in particular the energy to being on of the two quarks to infinity is not finite. At some point, when you have invested enough energy, there will be a spontaneous creation of a new quark-antiquark pair and you will end up with two bound quark systems, but no free quark. In this sense, one might say that the strong force is "infinitely stronger", but crucially this is not the aspect of the strong force that keeps nuclei together; the theory of pions shows no confinement.
"Does the strong nuclear force balance the electrostatic repulsions
between the protons or does it overcome the repulsion?"
This question is not worded very well. To explain that, let us consider the simplest systems containing just two neutrons or two protons. These systems do not have bound states. That means that the reason two protons do not have a bound state is not because the protons repel each other electromagnetically (remember that strong forces are pretty much the same for protons and neutrons, so if there were a bound state for two protons, there would also be a bound state for two neutrons), but because the strong force is spin-dependent, and two protons or two neutrons cannot have the same spin projections because of the Pauli principle, and the strong force between two nucleons with the opposite projections of spins does not enable a bound state. Remember that the system containing one proton and one neutron does have a bound state (deuteron) with the same spin projections for the neutron and the proton, as these two particles are distinguishable, and the Pauli principle does not extend to their system. Let me just add that at the distance of about 1 fermi characteristic for the strong force the electromagnetic energy for two nucleons can be orders of magnitude smaller than the energy of their strong interaction.
Best Answer
You answered to your own question, I believe.
The electrostatic force has an infinite range and it falls down with distance in an inverse square power law
$$ F_e \propto \frac{1}{r^2} $$
while the strong nuclear force behaves quite differently as it exhibits a property called confinement.
Confinement is still to be understood (it is related to one of the millennium prizes), but heuristically you can think of it as a force (for example acting on a pair of quark and anti-quark) that grows linearly with distance
$$F_s \propto r$$
just like a stretched spring. Although quite stronger than the electrostatic force, it happens that $r$ cannot stretch ad infinitum as the spring breaks into two pieces
and as such there is somehow a maximum distance after which the strong force does not attract any more.
As such nuclei, as you pointed out, will not be stable after some size as the electrostatic force will be the predominant force, which is repelling protons from each other.
Other references: http://en.wikipedia.org/wiki/Strong_interaction http://en.wikipedia.org/wiki/Millennium_Prize_Problems#Yang.E2.80.93Mills_existence_and_mass_gap