If you only have access to a 3DS, it is important that you update your firmware to 1.19 before you update the firmware on your 3DS. If you have access to a DS phat/lite/i/XL then you can update on any of those devices without worry as long as you have the latest EOS on the DSTWO.
Firmware 1.09 - works on 3DS's on v1.1.0-1 and prior
Firmware 1.12 - works on 3DS's on v2.2.0-4 and prior
Firmware 1.13 - works on 3DS's on v3.0.0-6 and prior
Firmware 1.15 - works on 3DS's on v4.1.0-8 and prior
Firmware 1.16 - works on 3DS's on v4.2.0-9 and prior
Firmware 1.17 - works on 3DS's on v4.3.0-10 and prior
Firmware 1.18 - works on 3DS's on v4.4.0-10 and prior
Firmware 1.19 - works on 3DS's on v4.5.0-10 and prior
1. Download firmware 1.19.
2. Extract "dstwoupdate.dat" to the root of your microsd card.
3. Put your microsd card into your DSTwo and insert it into a DS phat/lite/i/XL or your 3DS.
4. Turn on the DS and start the DSTwo, once you get to the plugin menu it should show a popup saying "Update system *your current firmware here* - 1.19", press A here to start updating. Make sure that you do NOT turn off or close the DS until it is finished upgrading.
5. If you did not see the popup, trying renaming "dstwoupdate.dat" to "ds2update.dat", then try it again.
You should now be on firmware 1.19, and dstwoupdate.dat (or ds2update.dat) should no longer be on your MicroSD card if it updated correctly.