Like Dan, I prefer a threaded mandrel when making stoppers using stainless steel cones. I drill and tap the blank to match the 3/6x16tpi thread.
But if you are making stoppers using the silicone sleeves that fit over dowels, the hole in the turning needs to be a full 3/8" diameter, and that's not compatible with the 3/8" threaded stud on the mandrel.
The dowels are actually tapered - the 3/8" end is glued into the finished turning, and the end that is covered with silicone is typically 1/2" diameter. Some options include:
1. Glue the dowel into the blank before turning, and then use a Jacobs chuck with a drawbar in the headstock to grip the 1/2" end of the dowel. The downside here is that Jacobs chucks grip in three places and may damage the dowel. Usually that damage will be hidden when the silicone sleeve is later slipped over the dowel.
2. Glue in the dowel, and then grip the 1/2" end with #1 jaws on a scroll chuck. Marginally better than using a Jacobs chuck because you don't need the drawbar and the chuck will grip the dowel at four places.
3. Glue in the dowel, and then grip the 1/2" end using a shop-made collet mounted in your scroll chuck. Make the collet from a scrap of face-grain wood that you have turned to form a tenon that is 3/4 - 1" in diameter. Then, with that tenon mounted in your scroll chuck, drill a 1/2" hole all the way through the tenonl. Finally, cut a saw curf on one side. Grip the split tenon in the scroll chuck with the saw curf between two jaws - so that when you tighten the jaws, they will squeeze the tenon and try to close down that saw curf, which also marginally reduces the diameter of the hole. Simply slide the blank on its 1/2" dowel into the collet, tighten the chuck jaws, and proceed with turning.