We each mod our Morrowind to suit our needs and tastes. More often than not, we look for mods that will balance the game, remove exploits and make us feel less like a cheat (especially once we know where the good stuff is).
Quite a few dedicated modders have overhauled the game greatly to achieve just that but I prefer to use a handful of mods only (less compatibility issues and greater choice).
I'll be listing those overhauls for those who like a one-stop-shop as well as my must have mods for a balanced game.

Overhauls

From my experience, playing with an overhaul, I tend to not even register the differences, sure, I'll make less money selling my ebony helmet, OK, my chitin armour gives me slightly better defense, clearly, Creeper is not as rich, etc... but it doesn't fundamentally change the way I play.

BTB edits An older overhaul that was very popular (I remember Telvani guard going bare chest because the Bonemold armour defense had been lowered)
Morrowind Rebirth changes everything from cities and its traders to the value of items, the defence of armour and the landscape. Generally thought to make the game more challenging because money is scarce(r), adds new creature variants and new dungeons (my favourite thing about Rebirth).
MNEM Good overhaul, especially since it is modular and comes with an installer and clear documentation. Carefully read the description to see the mods that have been merged such as Graphic Herbalism (outdated now, grab the MWSE version instead), thief experience overhaul, older versions of MGE and MCP are included (but optional).
Morrowind - Hardcore mod Now compatible with Patch for Purists. A great many changes to values, stats, etc to make the game more challenging.
Morrowind Fullrest repack Fullrest, the Russian modding website, overhauls everything including graphics. Never tested it myself, will add more info as I get some!
Morrowind Balanced Overhaul I have not tested this one personally but seems very detailed, also modular.

Like I said, one-stop-shop but with all of these, you want to check for compatibility.

My list

Patch for purists Fixes loads but stays true to the game.
Morrowind Code Patch Specifically potions and enchantment patches.
Harder Barter Unless your mercantile is high, you won't get rich quick (even if you know where to find that sweet daedric sword!)
Equipment requirements Won't let you use gear (efficiently) unless your skill is high enough. You may know where to steal that glass armour but it won't be of much use until your light armour is good enough.
Higher faction requirements At level 1, you will not be able to join every faction (and empty that equipment chest) and skill requirements for advancements are stricter too.
Antiwitcher by Archimag limits how much and how fast you can drink potions and eat ingredients. Your ability to drink potions improves with your attribute (Endurence and Speed) as well as with your alchemy.
Dynamic Difficulty by NullCascade where the difficulty of the game goes up as you level up (configurable in-game).

I find that with those, it doesn't matter that I get a glass cuirass at level 1, I can't use it or sell it (for much), stealing soul gems from Galbedir won't give me 80k septims, Creeper still has his 5k, I can't spam spells from rings and amulets, nor can I drink x potions while the game is paused.

Did I miss something?

Let me know what mods you like to use for a balanced game, I'm easy to find on Discord, twitter, twitch and YouTube, it's all DanaePlays!

I have not added realism mods because I feel that not having to eat, for example, doesn't make the game unbalanced. I have put together a mod list for role-playing that has essential mods, crafting mods as well as realism mods.
If you want mod suggestions for a more challenging game, head on right there.