You can’t do either. If you cap it you will create an “S” trap and will have siphonage problems. And if you put the P-trap in the basement your vertical line will be too long and velocity will cause the same problem. Both of these will leave you with permanent odor problems. And if you add a p-trap in the basement, the sink will be air bound due to the Studor vent and will not drain.
Hey there m&m ! Throw away that silly Studor Vent, a real plumber would not need these as we are educated enough to know the proper procedure for island venting, flat venting and the like. Because of the different variables involved I am not going to give you anymore advice. You should definitly get a plumber to do this for you, and depending on your local codes may need approval from the authority having jurisdiction. Good luck