For simplicity take the bosonic string theory in 26 dimensions. When you quantise the open and closed string you find excitations (states) of the string at any level $N$ with masses
\begin{equation}
M^2_\mathrm{open}=\frac{1}{\alpha '}\left(N-1\right),\qquad M^2_\mathrm{closed}=\frac{4}{\alpha '}\left(N-1\right).
\end{equation}
As $N$ takes on any non-negative value, you see that there is indeed an infinite tower of states. The tachyonic state (negative mass) at $N=0$ is a sickness of the purely bosonic string, which goes away for the superstring. At level $N=1$ you find the massless excitations (in 26 dimensions), and from the next level all states are massive. The mass is determined by the pre-factor. You can regard $\alpha '$ as the only free parameter of string theory, and it relates to the tension, $T$, of the string, which is expected to be slightly below the Planck scale, so that
\begin{equation}
\frac{1}{\alpha '}= 2\pi T \lesssim M_\mathrm{Pl}^2 = \left(10^{19} \, \mathrm{GeV} \right)^2 .
\end{equation}
So in conclusion, the infinite tower of massive excited states have masses at the order of the Planck scale, which means they are unobservable, and there is only a finite number of massless excitations. The same thing goes for the superstring.