Generally I’d say: Multiclassing is somewhat good in 5e, even though it’s an optional system and in that regard not necessary balanced, if you want to nerf it as a DM, here are some house rules, you could use:
1.) Favored class: You could simply say, that each (sub)race can only multiclass as long one of those classes is their favored class, saying that some races are too predestined to follow a certain path and can only break out if following that path truthfully. If we take the 3e favored class as a basic, it’d be:
|Race||Favored Class 3.5e||My suggestion|
|Elf, Dark||female Cleric, male Wizard||Sorcerer*|
*In a Forgotten Realms Setting (which I’ll play for some time now) I’d go with the same as 3,5e
2.) Even class-levels: A player can only have his class levels at a one level difference, so it won’t be possible to get only some levels in a class for features. If you choose this house-rule, you should grant your player an another Ability Score Improvement on level 10/10 or 7/7/6. Just to be sure, that there is not too much punishment.
3.) No further class-levels after taking another class: The character actually stops learning his old class and only can advance on his new class. Would be somewhat restrictive and makes a bit sense, but might not be restrictive enough, depending on your liking.
4.) Multiclass as Downtime Activity: Only between adventures a character can have the focus to actually attain a new class. He needs a teacher (or enough means to learn by himself) and time to get his studies done. I’d go with 1 gp per day and this table:
|Class||Time||Special Requirements Example|
|Barbarian||150 days||All days spent in the wilderness while battling at least four times a week or seeking spiritual contact under the guidance of a mentor|
|Bard||250 days||Getting trained by an experienced bard|
|Cleric||500 days||Learning in a temple, cloister, etc. of your god|
|Druid||500 days||Become a disciple of a great druid|
|Fighter||250 days||Being administered and studying in the military, a fighting school or under a private mentor|
|Monk||500 days||Getting trained by an experienced monk|
|Paladin||500 days||Purifying mind and body each day from all evil thoughts, seeking enlightment by the power of good|
|Ranger||250 days||All days spent in the wilds while studying under a mentor|
|Rogue||250 days||Getting training by a experienced rogue|
|Sorcerer||150 days||Being infused with one source of power fitting the origin (bathing in dragonblood, being in the Elemental Chaos without any protection, etc.)|
|Warlock||150 days||Met Otherwordly Patron already at the campaign one way or another|
|Wizard||500 days||A master to learn from and access to a magical laboratory|
You can forfeit any proficiency you’d normally get by multi-classing to reduce the time by 50 days per not taken proficiency. So if you have already some proficiency (especially armor and weapon) you can shorten the time without any penalty. You can learn not taken proficiencies at another downtime, 50 days and gp per proficiency to put the finishing touches to your training.
If you already have the spellcasting feature, you subtract 100 days of time, since you already now the basics.
You need a minimum of 50 days at least.
The best way is to actually talk to your player beforehand. Sometimes you might be too strict and maybe there might be a good idea behind it. This and of course the power of imagination brings us to the following!
Variant: Starting as a 0,5/0,5 Character
For those players who already want to start as mixed characters (to explain how they got to that point), I present a house rule here: Just start as half levels! Here comes how it works, we just take two classes and make those steps:
1.) You get only the Proficiencies of your Background and those in the Multiclasses Proficiency table (p.164)
2.) If you’re a spellcaster at 1st level, you only get half of known/prepared spells, spell slots and cantrips (rounded down)
3.) You get only those features in the following table
|Cleric||Spellcasting (half), Divine Domain (without benefits)|
|Druid||Spellcasting (half), Druidic|
|Ranger||Natural Explorer or Favored Enemy|
|Rogue||Sneak Attack, Thieve’s Cant|
|Sorcerer||Spellcasting (half), Sorcerous Origin (without any benefits)|
4.) Your starting hit points are the mid point of your both classes starting hit points (so a barbarian/fighter would have [12+10]/2 + CON-mod hit points)
5.) Take the mid point of the starting wealth of both classes to buy your start equipment.
6.) After getting level 2, you’re automatically upgraded to 1/1 and choose one of your classes, get all the left over proficiencies (saves, weapons and armors, etc.) and adjust your hit points as if you started with the chosen class and then took the second level at the other.
Alternatively you could talk to your DM and get a mix of proficiencies from both classes at the start, but since this opens up some PG-options, I didn’t took them in this section. Better let the DM decide of some of them depending on circumstances, so we don’t get a character with the best save proficiencies possible with those 2 classes, while shaving off some less prominent saves (here one proficiency should always lies in DEX, CON or WIS and the other at STR, INT or CHA).