Yes, the wavefunction will peak near the boundary of the forbidden region and this effect will increase at higher energy levels. In the limit of very high energy levels the quantum harmonic oscillator must reproduce the classical result and a classical harmonic oscillator will more likely be found near the endpoints of its motion since that is when it is moving much slower than in the center where it has maximum velocity and maximum kinetic energy.
A quote from Wikipedia:
Note that the ground state probability density is concentrated at the
origin. This means the particle spends most of its time at the bottom
of the potential well, as we would expect for a state with little
energy. As the energy increases, the probability density becomes
concentrated at the classical "turning points", where the state's
energy coincides with the potential energy. This is consistent with
the classical harmonic oscillator, in which the particle spends most
of its time (and is therefore most likely to be found) at the turning
points, where it is the slowest. The correspondence principle is thus
satisfied
The Wikipedia article also has the following animated images showing the wavefunction for eigenstates as well as animations for wavefunctions of states that are not eigenstates that begin to approximate the classical behavior of moving back and forth from one limit state to the other.
Some trajectories of a harmonic
oscillator according to Newton's laws of classical mechanics (A-B),
and according to the Schrödinger equation of quantum mechanics (C-H).
In (A-B), the particle (represented as a ball attached to a spring)
oscillates back and forth. In (C-H), some solutions to the Schrödinger
Equation are shown, where the horizontal axis is position, and the
vertical axis is the real part (blue) or imaginary part (red) of the
wavefunction. (C,D,E,F), but not (G,H), are energy eigenstates. (H) is
a coherent state, a quantum state which approximates the classical
trajectory.
