If you have a lathe and are practice in turning pens, moving to bottle stoppers should be simple.
There are many workable approaches, and every turner has his own preferences. Here are mine.
1. While you can do stoppers using a totally DIY approach, I prefer to purchase the actual stopper mechanism. I stainless steel for stoppers - in my opinion, the best are those sold by Ruth Niles. They come in a variety of forms - I prefer the design that has a 3/6"x16tip stud to screw into the turning. But I also make stoppers using the less expensive silicone sleeves that fit over dowels. I buy those from PSI, but there are other vendors.
2. I start by placing the blank between centers, turn to a cylinder, and put a tenon on what will eventually be the top end. I mount the blank in a scroll chuck. I use a spindle gouge to put a slight concave in the bottom - this concave needs to be about 1/16" deep - it's purpose is to recess the top of the metal/silicone into the turning so that the join between the stopper base and turning isn't visible from the side. This makes the finished product look a bit neater. It's also possible to create the same effect by drilling a rabbet with a forstner bit. Then, I drill a 5/16" hole in the center of that recess. I do the drilling on the lathe so that the hole is absolutely centered on the recess or rabbet. I use a 3/8"x16 plug tap to thread this hole. Finally, I sand the bottom and apply a coat of WOP to seal the bottom of the turning. The main objective is to prevent wine from seeping into the wood - red wine in particular could stain the turning.
3. Then, I remount the blank on a stopper mandrel. I purchased my mandrel from PSI, but there are other vendors (including Ruth Niles), and it's a simple matter to make one using a scrap of wood and a 3/8"x16tpi bolt. Then, I turn whatever shape it want. A key point is that while having a pointed top might look nice, a stopper with a pointed top isn't very practical - having a flat or rounded top makes it much easier to put the stopper into a bottle. Finally, I sand the turning and apply finish. Normally, my preferred finish is WOP because it produces a tough finish that looks nice and that is resistant to alcohol. However, I have made a few stoppers with captive rings, and to simplify the finishing process on those, I've used pure tung oil.
4. Finally, assemble the turned top and the base. With the studded-metal base, all that you have to do is screw the top onto the base. If the threads in the turning are punky, you can put a drop or two of thick CA, polyurethane glue, or epoxy to assure that the top won't come off. With the silicone sleeve/dowel design, the dowel is too large to fit naturally into the threaded hole - options include reaming out the hole in the turning (put a 3/8" bit in a hand drill and, while handholding the turning, enlarge the hole), threading the dowel using a 3/6"x16 die (doesn't work really well - DHMHIKT), or mounting the dowel in your lathe (using either a scroll chuck, Jacobs chuck or a collet chuck), and reducing the diameter to 5/16" with a skew or even sandpaper. I usually glue the dowel into the turning using PVA glue. Often, it is possible to remove the sleeve from the dowel, and in that case, I will typically apply a small bead of a silicone adhesive around the top of the sleeve to prevent wine from seeping past the sleeve and staining the wooden dowel.
Some people prefer to reinforce the threaded hole in the turning with thin CA glue, and after it cures, run the tap into the hole again to clean up the threads. I've done that, but I've come to the conclusion that its not necessary. The threads are used twice - once to mount the blank on the mandrel, and again to mount the turning on the threaded base. After the stopper has been assembled, it never needs to come apart again. My experience is that unless the wood is really soft (eg, spalted and then some), the effort required to reinforce the threads isn't necessary.
