You have quite a few options, but some will obviously be undesirable or impossible to do because of your situation.
First, I'll assume that trailing spaces in the data are because the data types for the source columns are CHAR or NCHAR. You can change the data types in the source database to VARCHAR or NVARCHAR. This probably isn't a good idea.
If the data types in the source data are VARCHAR or NVARCHAR and the trailing spaces are in the data, you can update the data to remove the trailing spaces. This is probably not appealing either.
So, you have SSIS and the best place to handle this is in the data flow. Unfortunately, you must develop a solution for each column that has the trailing spaces. I don't think you'll find a quick and simple "fix all columns" solution.
You can do the data trimming with a script transformation, but you must write the code to do the work. Or, you can use a Derived Column transformation component. In the Derived Column transformation you would add a derived column for each column that needs trimming. For example, you would have a firstname column and a lastname column. The derived column value would replace the existing column value.
In the Derived Column transformation you would use SSIS expression syntax to trim the data. The firstname and lastname trim expressions would be
RTRIM(firstname)
RTRIM(lastname)
Performance will probably be better for the Derived Column transformation, but it may not differ much from the script solution. However, the Derived Column transformation will probably be easier to read and understand later.