Windows Authentication In SQL DTS packages
You can do this by giving the specified Windows account the necessary permissions to the servers/databases that the SSIS job connects to, and then running the DTS package under that Windows account.

For example, to run a SQL Agent job step with the Windows account, you have to create a SQL Server credential, then a SQL Server Agent proxy for that credential, and then assign the job step that runs the package to run under the proxy.  I don't recommend it, but for completeness will mention that if you are using a domain account for the SQL Server Agent account, you can simply give permissions to that account, and have the job step run with it.  The problem with this is that you have one account with permissions to numerous objects and it becomes a security concern.

If you are using a Windows Scheduled Task, you need to configure that task to run with the specified account.

You'll also need to change the connection strings to remove the SQL login and password, and specify ;Integrated Security=SSPI so that it will use Windows authentication.

This thread also mentions that a linked server can be used, but doesn't provide any further details.