If you want to use the new features of the reminder app updated under iOS 13, think about it first: compatibility with old devices will be removed.
More precisely: the reminders can no longer be synchronized via the iCloud if it is a device that uses older software than iOS 13 or macOS Catalina. The attentive reader will have noticed that there are currently no devices besides the iPhone to which reminders can be synchronized. iPadOS will only be available in a few days and Catalina for the Mac is not yet available.
The settings dialog kindly asks if you want to switch to the new version. It’s doubtful if it goes back again.
Also note: if you frequently use the app reminders and especially the synchronizations possible via the iCloud, you should determine before switching if you have a device in use that runs on iOS 12 or macOS 10.13 High Sierra. Especially the latter should be common, since it affects many devices from 2010/2011, which will be provided with software updates for more than a year.