Preheat the oven to 350F (180C) degrees and place the oven rack in the bottom ⅓ of the oven. Wrap the outside of a 9-inch springform pan with aluminum foil, covering all seams and ensuring no areas are exposed.
In a medium bowl, combine gingersnap crumbs, brown sugar, and melted butter.
Press the mixture into the bottom of the prepared pan. Bake in the preheated oven for 8 minutes. Remove from the oven, leaving it on.
In a large bowl, beat the softened cream cheese until soft. Scrape down the sides of the bowl as needed.
Beat in sugar, cornstarch, and nutmeg. With the mixer on low speed, mix in vanilla extract, rum/rum extract, and eggnog. Beat in the eggs until just combined. Place the cheesecake crust in the center of a large roasting pan and pour the cheesecake filling on top.
Pour hot water into the roasting pan until there's about ½-1 inch of water in the bottom. Place the roasting pan with the cheesecake in the oven and bake for 50-60 minutes, or until the top is almost set, except in the very middle. Remove from the oven. Cool the cheesecake in the water bath until it reaches room temperature.
Remove the springform pan from the roasting pan, cover the cheesecake, and refrigerate for at least 6 hours. When ready to serve, trace around the edges of the cheesecake with a thin knife. Unclamp the springform pan.