I would suggest that you think about meeting your requirements with a slightly different approach.
MapServer can cascade WMS services, but I am not aware of any public WMS services that serve up OSM data for free. Most OSM data services that I am aware of use the tiled TMS specification.
If your version of MapServer is compiled with a version of GDAL that supports the TMS standard (via the WMS mini-driver) http://www.gdal.org/frmt_wms.html, you can theoretically pull it off, but I don't believe that it is widely used.
The simplest solution is to keep the basemap and data layers separate on the server side and combine them in the client. Configure your map client to use OSM data tiles from MapQuest Open, CloudMade, OSM, or someone else for the base layer. Create a transparent layer for your own data and pull it into your client via WMS.
Make sure that you look at and honor the license/usage requirements for your source of OSM tiles.