There is no support for style information in shapefile format. However, you can store the style info as a regular attribute value. Using OGR's SQL you can extract the style string from the kml data and then store it in the shapefile. Please go through the OGR SQL page. You can get the style from the special field named "OGR_STYLE".
First, make sure you are using the libkml driver and not the kml driver.
The libkml driver can read the style information. Then you can get that
information in your shapefile by specifically requesting the ogr_style