When calculating a precipitation index, should it be constrained to a basin?
Normally you could calculate a precipitation index for any location with a long-term record.  A basin will have nearly the same index, regardless of which station location you use, so it can be appropriate to just calculate a single value for the whole basin, as you've suggested.  The project scope will determine what level of spatial variability you need, and sometimes political boundaries are good enough (e.g. counties). Applying this same idea to a satellite-derived product, you could use a simple ecoregion boundary (see here) to determine the spatial resolution.  You may also want to determine spatial boundaries baed on an environmental condition that can interfere with the accuracy of the satellite retrievals you are using (e.g. surface albedo or elevation).
I'm intrigued by the prospect of calculating a precipitation index from satellite imagery (instead of ground stations) and wonder which instrument(s)/products you are planning to use, especially since the calculation involves a climatological component that is difficult to build without continuous records. I suppose it is possible to train your algorithm to give results consistent with ground stations, otherwise I would worry about accuracy.  Good luck!