Before updating plugins a production server you should make a backup of your entire site. There are a wide variety of options here. You could simply grab a backup of your database and FTP your files to a backup location. A better option is to use a backup plugin or service. These make it much easier and faster to make a backup, which makes it much more likely that you’ll actually do it.
WordPress.org has many many backup plugins available. They have a variety of options and tools, feel free to experiment and find one that fits you.
BackupBuddy is a plugin with an optional storage service that has a great many options and comes with paid support.
VaultPress is both a plugin and a storage service and is owned by Automattic.
Regardless of method, please be sure to always make a backup of your entire site before updating any plugins.