How does causality (i.e. unit step) affect the DTFT of a sine or cosine wave? Tables of common Discrete-Time Fourier Transform pairs list the transform of a sine wave:

$ \sin(\omega_0\ n) $ and its transform:
$ -j\pi\ [d( \omega\ - \omega_0\ ) - d( \omega\ + \omega_0\ )] $

And the cosine: 

$ \cos(\omega_0\ n) $ and its transform: $ \pi\ [d( \omega\ - \omega_0\ ) + d( \omega\ + \omega_0\ )] $



How might the results differ if the sin or cosine is causal? That is, I would like to determine the Fourier Transform of the following (sin or cosine multiplied by the unit step):

$ \sin(\omega_0\ n) u[n] $ or  $ \cos(\omega_0\ n) u[n] $

So regarding the DTFT of the signals above, is the only method to realize that multiplication in the time domain is convolution in the frequency domain? Or is there a more simple rule-of-thumb, or property, of the Fourier Transform that I am missing?

(This is not homework, but it is for studying, so it would be helpful to include resources along with answers.)
Multiplication in the time domain is convolution in the frequency domain. The frequency response of a unit step has 1/frequency shape and it's pretty messy. 

What you will see is that the overall frequency response is not a delta impulse anymore but that the lines widens and that the get "skirts". These will bigger and more "wiggly" for the cosine than for the sine as the step function slices right through the maximum of the cosine.