Just follow
this and you should be good to go.
The OP said he doesn't want CFW, so 3ds.guide doesn't really help.
It's been awhile and have totally lost touch with the scene. Basically I'm having an old 3ds running 8.1.0 FW with Gateway 3.0.4 and am thinking of "updating". Would it be possible to stick with my current 8.1.0 and load in Gateway 3.7.2 and above still using the browser hax? If so what do I need to do? I'm trying not to go into using CFW.
Thanks!
Updating your Gateway should work just fine (and you can just downgrade it if it doesn't work), but there's no point as newer games can't run on firmwares like 8.1 at all, meaning you will have to update. However, updating would get rid of all the patched exploit entrypoints (eg: browserhax). I THINK it's possible to keep exploits like browserhax though if you find a game cartridge that comes with an update that doesn't update the browser (can't remember the specifics, someone will have to chip in here for this).
If you want to play the newer games though but keep using Gateway you can do something like EmuNAND 11.2/SysNAND 11.6, but it's just best to throw away that flashcart and simply use CFW nice and cleanly. Less of a mess, no worries about updating, unpatchable access to the HomeBrew Launcher (meaning you never lose homebrews), ability to play all the newest games just fine and generally better as partitioning your SD card will probably make it at least slightly slower overall, on top of not having the EmuNAND take up precious space in your SD card. And if you ever want to go back, it's very simple to uninstall CFW and it's almost fully automated so you run no risks at all.