I had the same problem in the SSIS 2008 data flow package . I had a CSV (comma delimited) source data with one of the columns data quoted with a double quote( " ). The import worked fine using the SQL 2000 DTS import as it recognized the Text Qualifier value as a Double Quote {"}, but this was not the case when I started to use the SSIS 2008. The Text Qualifier shows <None> on the Flat file connection manager object (Text Qualifier) property. I tried to change the <None> to what I have in the DTS 2000 Double Quote {"}, but it didn't fix the problem. I kept researching on this until I found the simple answer to fix this is to put only (") in the Text Qualifier Property and this worked fine and I got rid of the double quotes in the data and the data imported successfully as it was in the source even with the data which contained comma in between.
You find below the link where I found my answer from: