What is wrong with a nonrenormalizable theory?

In the modern effective field theory point of view, there's nothing wrong with non-renormalizable theories. In fact, one may prefer a non-renormalizable theory inasmuch they tell you the point at which they fail (the energy cut-off).

To be concrete, consider an effective lagrangian expanded in inverse powers of the energy cut-off $\Lambda$:

\begin{equation} \mathcal{L}_\mathrm{eff}(\Lambda)=\mathcal{L}_\mathrm{renorm}+ \sum_\mathcal{\alpha}\frac{g_\alpha}{\Lambda^{ \operatorname{dim}\mathcal{O}_\alpha-4}}\mathcal{O}_\alpha \end{equation}

where $\mathcal{L}_\mathrm{renorm}$ doesn't depend on $\Lambda$, $\mathcal{O}_\alpha$ are non-renormalizable operators (dim. > 4) and $g_\alpha$ are the corresponding coupling constants. So at very low energies $E\ll \Lambda$ the contributions from the non-renormalizable operators will be supressed by powers of $E/\Lambda$.

That's why the Standard Model is renormalizable, we're just unable to see the non-renormalizable terms because we're looking at too low energies.

Notice also that as we increase the energy, the first operators to become important will be the ones with the lower dimension. In general, contributions from non-renormalizable operators will become important in order given by their dimension. So you can see that, although there are infinite possible non-renormalizable coupling constants, you can make the approximation of cutting the expansion of the effective lagrangian at some power of the cut-off and get a finite number of parameters.