What's the meaning of wavefunction being finite?
A broader point regarding functions that are finite:
A function is finite if it never asigns infinity to any element in its domain. Note that this is different than bounded as $f(x):\mathbb R \to \mathbb R \cup\{\infty\}: f(x)=x^2$ is not bounded since $\lim_{x \to \infty}=\infty$. However, $f$ is finite since it does not assign $\infty$ to any real number.
This is a generalisation of the comments written above, not specifically quantum related, but a more general point on mathematical functions.