Unfortunately SSIS in SQL 2005 and 2008 does not support embedded text qualifiers in CSV files.

Vote on it at Microsoft Connect.

https://connect.microsoft.com/SQLServer/feedback/ViewFeedback.aspx?FeedbackID=312164

One workaround is to read in a single line at a time, then parse each line in a Script component before going to the destination.

[Source] -> [Script Component] -> [Destination]