What happens to an SSIS package when I alter a table that is used as a destination?
If you aren't changing the destination table name or the name or properties of any existing columns, then the package may continue to run without any changes.

If you need to actually include these new columns in your import, though, then you will probably need to update the package.

If you don't have the dtsx file, then I assume the package is published in an SSISDB?

Here are some articles that discuss editing/exporting a published package for different SQL/SSIS versions:


How to export a package from SSISDB?
Editing Published SSIS Package in SQL Server 2012.