You need to download the available satellite data of the whole basin, and then use the tool "extract by mask" in ArcMap with your mask being the sub basin to get the data for the sub basin.
I am assuming that you need a time series of your sub-basin and perhaps, with the areal mean. You could use the xarray and seaborn packages in python. All you need is a the shape file of your sub-basin and the gridded dataset of your bigger basin. Then, use your shape file to mask out your sub-basin from the bigger dataset. You may decide to leave your output as netcdf series or an areal timeseries.