I like the house rule that someone suggested earlier, that "all class levels must be within 1 level of each other." This forces characters to multiclass by 2nd level if they are going to, and prevents cherry-picking or level-dips. But it might be a bit too restrictive.
Then, if you were to bring back the "Favored class" mechanic from earlier editions, you could waive that restriction for a race's favored class (and waive the ability score requirements for that class in the PHB.) Like, a High Elf Wizard 10/Monk 1 would be fine, since Wizard is the High Elf's favored class...but a High Elf Bard 10/Monk 1 wouldn't be. If you were a high elf Fighter, you wouldn't need to have an Int of 13 to take levels of wizard, but you probably would want to anyway.
The issue of "what to do about humans?" would keep coming up, though, and I'd be inclined to set it up so that humans don't have a favored class. They are already a stats-and-options smorgasbord as it is...they don't need extra help in the Multiclassing Department.
Maybe something like this?
Race (Favored Class)
Aasimar (Paladin or Warlock)
Arakocra (Fighter)
Bugbear (Rogue)
Centaur (Druid or Ranger)
Changeling (Rogue)
Dragonborn (Sorcerer)
Dwarf (Fighter)
Elf, Drow (Cleric or Wizard)
Elf, High (Wizard)
Elf, Wood (Ranger)
Firbolg (Druid)
Genasi, Any (Sorcerer or Druid)
Gith (Psion? or Monk)
Goliath (Barbarian or Fighter)
Gnome, Forest (Druid)
Gnome, Rock (Artificer)
Goblin (Rogue)
Halfling, Lightfoot (Rogue)
Halfling, Strongheart (Fighter)
Human (no favored class)
Half-Elf (Bard)
Half-Orc (Barbarian)
Hobgoblin (Fighter)
Kalashar (Monk)
Kenku (Rogue)
Kobold (Rogue or Sorcerer)
Lizardfolk (Ranger)
Locathah (Fighter)
Loxodon (Cleric)
Minotaur (Barbarian)
Orc (Barbarian)
Orc, Eberron (Barbarian)
Shifter (Ranger)
Simic Hybrid (Druid)
Tabaxi (Rogue)
Tiefling (Warlock)
Tiefling, Feral (Warlock or Sorcerer)
Triton (Paladin)
Vedalkin (Monk)
Verdan (Psion? or Wizard)
Warforged (Fighter)
Yuan-ti Pureblood (Sorcerer)