You didn't say which version of SSIS/SQL Server you are using.
First, if you have the separator (in your case comma) as part of the data, you have to escape them. For example, like this:
where I have escaped the second column with double-quotes. In addition to that, you have to tell SSIS that you are using double-quotes as escape character.
Even with all of that, some versions of SQL Server cannot properly process csv files. See some discussion here. I don't know if the issue has been fixed in a newer version of SQL Server.
What I normally do is to ask the data provider to send the data using some other delimiter such as pipe (|) or caret (^) that is guaranteed to be not part of your data. Which delimiter you should use depends on the nature of your data and the business segment you are in. In my case - financial data - usually pipe character works.