A letter template must have a Data source of People to be used for birthday reminders. For SMS messages the Preferred sending method must be SMS/text message, and for e-mails it must be Any mail or E-mail. If these conditions aren't met, VSys won't show the template as an option here.
You can include the person's age (or not). If you do, though, use the field Age at nearest birthday rather than Age. This will bring up the "birthday" age since when you're sending these out, the person hasn't yet reached their new age.
Birthday greetings can also be sent automatically to the kiosk. From the Kiosk panel see Holiday Greetings.