The original Nintendo DS and DS Lite have no user-profile system, so you cannot store a birthday on the console itself.
Workarounds
System Date/Time: Set the correct date in System Settings → Calendar. The DS will show the date on the main menu, but won't celebrate birthdays.
DSi/3DS Calendar Apps: On DSi/DSi XL, download a calendar app from the DSi Shop (now closed) or use the built-in calendar on 3DS to add a birthday event with notifications.
Physical Reminder: Write it on a wall calendar or phone.
Only the Nintendo 3DS family and later consoles (Switch) support Mii profiles with birthday fields.
