Sextante is mainly a library to be used from a gis desktop application like gvSIG, qgis or ArcGIS.
If you want to use any of the algorithms of Sextante on postgis layer, just choose any a gis desktop app, load the postgis layer, and use Sextante on it, as if it was a shapefile.
For example gvSIG comes with Sextante an a lot of algorithms without the need to download anything more, but if you need to run the algorithms of another providers (grass, ...) from Sextante, probably your best choice is qgis.
For a server side solution, but you should implement the analysis that you need with the postgis procedures, or take a look to applications that implements WPS like the latest release of opengeosuite.