You can definitely control what columns will be exported to the output feature class. You should choose what fields will be added in the Field Map of Join Features (optional) section of the Spatial Join GP tool dialog (available in all licenses).
From the tool Help:
Controls what attribute fields will be in the output feature class.
The initial list contains all the fields from both the target features
and the join features. Fields can be added, deleted, renamed, or have
their properties changed. The selected fields from the target features
are transferred as is, but selected fields from the join features can
be aggregated by a merge rule.
You just right-click the field and choose whether you want to delete it.
Another in my view tidier alternative, is to use Make Feature Layer tool to have only those fields that you need before feeding the data into Spatial Join tool. You can create a layer in your map document which will have only fields you need and you can use this layer as input for your GP tools. This is particularly convenient when automating the process with ModelBuilder or Python when you work with feature classes but need to "filter" them first to avoid having unnecessary information in the output.