No, the outlook client does not have to be running to use send-mail in a dts package.
The outlook client needs to be installed on the server and the profile needs to be created for the user context that will be running the dts package.
Here is where the confusion typically lies. When you run a DTS package interactively, the security context is who-ever you are logged in as. As such, it is relatively easy to debug. What most people don't understand is that if the DTS package is running based on some automated feature (timer, or another application starting it) the DTS package typically runs under the SQL Agent ID (Check the logon-id of the user that starts SQL Agent in control panel Services). You will need to use a specific ID, rather than system. This is because to use SQL Mail, you will have to interactively log-on as that user and configure outlook. Once you have done that, the SQL Agent will be able to send and receive e-mail without you being logged in.
Good luck.