The modify stage can do this, in a manner. While it won't select all string columns from a schema, if you have a parameter called "ColumnName" with the name of the string column to trim, then the modify action of
"#ColumnName#=trim(#ColumnName#)" will do what you want for one column.
Sorry, I mis-pasted the function, it is as Ray notes, "string_trim". I took a quick look at what might be done that could apply across a whole schema but don't think that there is any such functionality.
You could perform the import (read) using a single VarChar column (which could be trimmed in a Modify stage) and then perform the parsing in a Column Import stage (which can use a schema file). An added benefit is that your parsing can be performed in parallel mode.
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