Sidecap module
There is a module that uses the sidecap package available from http://www.physics.utoronto.ca/~jrioux/resources/lyx/. Download the sidecap.module file and follow the instructions on the that site for installing.
In LyX, go to Document --> Settings --> Modules, and add the Sidecap module which now should be available. To add a figure with a side caption, simply choose Insert --> Float --> Figure (side caption). Similarly for tables.
Original answer -- without modules
I'm afraid sidecap doesn't change the behaviour of normal figures, it only defines new environments where the caption will be placed on the side. To use sidecap I think you will have to add these environments "manually", in ERTs.
After adding \usepackage{sidecap}, you can do the following where you want to add a figure:
- Add an ERT with Ctrl + L, and type
\begin{SCfigure}.
- Move the cursor outside the ERT, and add the image as you normally would, e.g. via the button on the toolbar.
Add a second ERT after the image, and type
\caption{<Your caption here>}
\end{SCfigure}
An example of how it may look inside LyX:

Cross referencing
If you want to add a cross reference to such a figure you have two options I think:
- In the second ERT, write
\label{yourlabel} after the \caption{}. The problem with this option, is that the label doesn't show up in the LyX's GUI, so you'll have to write in in manually.
- Write
\end{SCfigure} in a separate ERT, and in between this one and the one with \caption{}, add a label with the button on the toolbar or Insert --> Label.

For both cases, instead of writing the caption in an ERT, you could also add it using Insert --> Caption.