What is Imaginary in Fourier transform?
First of all, in order to have a Fourier transform, the original signal has to be multiplied by the unit step function:

$$x(t)=e^{-t}u(t)$$

Giving indeed the transform:

$$X(\omega)=\frac{1}{j\omega+1}$$

The way to plot a complex function on the frequency domain is by finding both its amplitude and its phase, and drawing one graph for each.

For the amplitude case, find $|X(\omega)|$, which is:

$$X(\omega)=\frac{1}{\sqrt{\omega^2+1}}$$