SUMMARY OF EDITED VERSION: You cannot place any conditions on $V(x)$ and $E$ that guarantee that solutions to the time-independent Schrödinger equation are normalizable, for something of a silly reason.
Initial, partial answer: If the potential is bounded below by some value $V_\text{min}$, then a solution to the time-independent Schrödinger equation with $E \leq V_\text{min}$ cannot be normalizable. Proof: suppose $\psi(x)$ is a normalized wavefunction that solves Schrödinger's equation with eigenvalue $E$. Then
$$
E = \langle \psi | H | \psi \rangle = \int \left[ \frac{\hbar^2}{2m} \left| \frac{d\psi}{dx} \right|^2 + V(x) |\psi|^2 \right] dx > \int V(x) |\psi|^2 dx \geq \int V_\text{min} |\psi|^2 dx = V_\text{min}.
$$
We have strict equality in the third step because $\psi$ cannot have zero derivative everywhere and still be square-integrable. (Note that this result is also a problem in Griffiths's text, with a different suggested method of solution.)
EDIT:
In fact, I think that it is impossible to place constraints on $V(x)$ and $E$ that guarantee a normalizable solution, for the following rather silly reason: If we view the time-independent Schrödinger equation as an ODE, then for any value of $E$ there are two linearly independent solutions to the second-order ODE
$$
- \frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \left[ V(x) - E \right] \psi = 0.
$$
So even if one square-integrable function $\psi_1(x)$ satisfies this equation, there will also be another linearly independent solution $\psi_2(x)$ that satisfies the equation as well, and this second solution will in general not be square-integrable (see below.) For example, if you try to solve the above ODE for the harmonic oscillator with $E = \hbar \omega/2$, you'll get two solutions, one of which has the usual $e^{-x^2/\sigma^2}$ behaviour (and is therefore square-integrable) and the other of which goes as $e^{x^2/\sigma^2}$ asymptotically and is therefore not square-integrable.
You might wonder whether it is possible that both $\psi_1(x)$ and $\psi_2(x)$ could both be square-integrable; but unfortunately this turns out not to be the case. To show that this can't happen, we can use logic like that of Ali Moh's answer. If the leading-order asymptotic behavior of $V(x)$ is proportional to $x^\alpha$, and the potential $V(x) \to \infty$ as $x \to \infty$, then we can write the Schrödinger equation asymptotically (after some rescaling) as
$$
-\psi'' + (\beta x^\alpha - e) \psi = 0.
$$
where $e$ is proportional to the energy and $\beta > 0$. If $\alpha > 0$, then the first term in the brackets will dominate, and the equation then has the approximate solution (via Mathematica)
$$
\psi(x) = \left\{ \sqrt{x} I_{-1/(2+\alpha)} \left( \frac{2 \sqrt{\beta}}{2 + \alpha} x^{1 + \alpha/2} \right), \sqrt{x} I_{1/(2+\alpha)} \left( \frac{2 \sqrt{\beta}}{2 + \alpha} x^{1 + \alpha/2} \right) \right\}.
$$
where $I_n(x)$ is a modified Bessel equation of the first kind. Now, if we have two solutions $\psi_1$ and $\psi_2$ that correspond to the same value of $E$ and are both square-integrable, then some linear combination of them should behave like each of these two solutions asymptotically; but both of these solutions diverge, and $\psi_1$ and $\psi_2$ both have to go to zero asymptotically to be square-integrable. Hence, one of the solutions $\psi_1$ and $\psi_2$ must not be square-integrable. (The same logic holds if $\beta < 0$; we just get ordinary Bessel functions instead, which are still not square-integrable.)
If, on the other hand, $\alpha \leq 0$, then the potential is bounded above and we recover the case discussed by Ali Moh. Again, the generic solution to the ODE will either be oscillatory or exponential, and so at most one of the linearly independent solutions to the time-dependent Schrödinger equation will be normalizable.
Beyond this, I'm not sure what exactly one could do. You might ask, "Can we put conditions on $V(x)$ and $E$ such that a normalizable solution to the Schrödinger equation exists?" But this is basically asking "Is $E$ in the spectrum of the Hamiltonian, when the Hamiltonian is acting only on the space of normalizable wavefunctions?" In other words, you're asking for the energy eigenstates and eigenvalues, which is what we're usually looking for anyway. You could always use variational-principle results to place bounds on the spectra of particular potentials; and there's the well-known result that at least one bound state ($E < 0$) exists for any attractive potential in 1D (and 2D). But I'm skeptical that a more general result exists.
