The dimensional formula for $E=E_0{e}^{-t/t_0}$ as  $e^{-t/t_0}$ is a dimensionless quantity.

In the question, $E$ and $E_0$ represents the energies. $E$ is the time-averaged energy of the oscillator which is dimensionally represented as $M{L}^2{T}^{-2}$ and $E_0$ is the total energy of an undamped oscillator which is represented by $E_0=\frac{1}{2}m{A}^2{w}_0^2$, where $m$ is the mass, $A$ is the amplitude, and $w$ is the angular velocity.

The dimensional representation of the total energy of an undamped oscillator is

$E_0=M{\left(L\right)}^2{\left(T^{-1}\right)}^2=M{L}^2{T}^{-2}$

Thus, the dimensional formula is $E=E_0{e}^{-t/t_0}$.