The short answer is, yes, the definitive screening design is appropriate for the case with multiple responses.
The considerations for using a DSD over other design methods have nothing to do with the number of responses. They have more to do with deciding if this is a screening situation. That is, do you expect the key screening principles to hold and is it not one of the cases for which DSD was not intended (see Bradley Jones blog post).
If you are referring to using the Fit Definitive Screening option to select the model, then the responses are fit individually. This analysis scheme is not possible with more than one response at a time. You could save the fitted model for each response and then combine them by selecting Graph > Profiler. You would be able to examine how each factor affects all of the responses. (It is generally considered a best practice to fit the best model individually to each response.)
The other design choices produce the Model table script. JMP will jointly fit the multiple responses by default, but you can over-ride that choice. Fitting a common model still determines individual parameter estimates but the model contains the same terms for every response. The Effect Summary presents the result for each term for the response which is most significant (smallest p-value).
You can analyze a DSD experiment using the same Fit Least Squares platform by starting with Analyze > Fit Model if you prefer that behavior. You can always take control and choose the other behavior if it suits your purpose better.
