Monk is always better than (or as good as) Barbarian for boosting AC.

That is, as long as you keep your Wisdom up. There's a spreadsheet with lots of numbers here. It includes some options other than Barbarian and Monk that vary from somewhat controversial to completely ridiculous. I'll break down those options, pros and cons of the different choices, and so on below.

Barbarian

The Barbarian's Unarmored Defense is 10 + Dex + Con, which means it's determined solely by the stats of what you're turning into. Your race and ability scores are up to you, as long as you have at least 13 Wisdom and 13 Strength. You also get Rage, giving you up to 2 minutes of resistance to bludgeoning, piercing, and slashing damage per day, as well as +2 damage on your attacks, depending whether your DM thinks your attacks are Strength-based.

Monk

The Monk's Unarmored Defense is 10 + Dex + Wis, meaning that to get the most out of it, you need to have a high Wisdom score. You'll also need to have at least 13 Dexterity, because of the Monk's multiclassing requirements. As a Druid, that also controls the effectiveness of your spells (when you use them), so it's nice to have it high anyway. The spreadsheet includes a Wisdom score in the calculations - the number that's there assumes you're using point-buy and keeping your Wisdom score as high as possible.

Draconic Sorcerer

This is the only other PHB option. It's somewhat controversial whether Wild Shape forms can benefit from Draconic Resilience's scaly skin, so don't expect this to fly at every table. It requires you to have at least 13 Charisma, but on the other hand, your spell slot progression doesn't lose a level. That said, I don't recommend this, because for AC it can only ever be as good as Monk, never better. Why? Because as long as you have 16 Wisdom or more, the Monk's 10 + Dex + Wis will always be at least 13 + Dex - the same as Draconic Resilience.

Lizardfolk

This isn't a 1-level multiclass, obviously, it's a race - from Volo's Guide to Monsters. It gives you the same AC calculation as the Draconic Sorcerer, as well as essentially the same controversy. On the upside, if you use this, you don't have to spend a level outside Druid.

Immortal Mystic

Mystics (from Unearthed Arcana) of the Order of the Immortal get the same AC calculation as the Barbarian. However, if they keep their psychic focus on the Iron Durability discipline, they get +1 to AC on top of that, for an overall AC of 11 + Dex + Con. This means they have a higher AC than Monks in some forms, although still not many. That said, being from Unearthed Arcana means that they're not official and they're explicitly not balanced for multiclassing.

Stone Sorcerer

Another Unearthed Arcana option, the Stone Sorcerer gives an AC of 13 + Con. This tends to beat out the Monk for forms with high Constitution and low Dexterity, which includes all of the forms from CR 4 upwards as well as (appropriately) the Earth Elemental. Their mechanic allows you to turn your skin stony after you Wild Shape, which should avoid the controversy of the Draconic Sorcerer, although it means that if you wait until combat starts to change forms you might lose a turn doing it. As with the Mystic, being from Unearthed Arcana means that they're not official and they're explicitly not balanced for multiclassing.

Tortles

As with the Lizardfolk, this one is a race. In the case of the Tortle, it's from the Tortle Package, which is an official pdf you can buy at DM's Guild. It lets you ignore all this calculation business and just have an AC of 17 all the time, which is both extremely convenient and incredibly powerful, toppling the Monk as the overall best option. When you add in that it doesn't involve multiclassing, and has a Wisdom bonus to boot, it becomes downright ludicrous. That said, you'll have to persuade your DM that all your Wild Shape forms have a giant turtle shell. To be honest, I'd be amazed if this flies at any table.