I'm asking as a player. We are at a part of the campaign where the enemy wizard/sorcerer always has dispel magic prepared. As a support wizard, I find that the enemy caster always dispel the buff I've cast on my frontline fighter, or the debuff on their ally. This is not about whether the DM is playing fair or not.

Is there any method that can protect an enchantment or spell from dispel magic? I'm fine if it's only temporary, but at least I need to make the spell very hard to dispel, at least for a few rounds to make it not immediately useless after the caster's turn.

I'm fine with increasing the dispel DC, giving a one-time or one-round protection from dispel magic. Using magic item or feat/classfeat is also welcome.

To guarantee it is not dispelled,

Globe of Invulnerability

Any spell of 5th level or lower cast from outside the
barrier can't affect creatures or objects within it, even
if the spell is cast using a higher level spell slot.

You might need to coordinate with another caster if your buff also requires concentration. However, this Globe will protect your ally from getting dispelled, regardless of how strong the Dispel Magic cast by your enemy is. Seconded by Sage Advice:

• Can dispel magic end globe of invulnerability?


• Yes, dispel
  magic can dispel the barrier created by globe of invulnerability,
  but not any magical effects that are active inside
  the barrier.

Caveats:

• Since the sphere has a range of self, the buffed person must either be able to cast the sphere, or stand near someone who can.


• While enemies can dispel the Globe, you can keep casting it and protecting the buff.


• Also, a Wish spell to dispel your buff will bypass the Globe. But it should be a good trade-off for you if the enemies do use a Wish just for that buff.

Other ways of helping, in addition to other answers (upcasting your buff, controlling enemies, counterspelling their dispels, or hiding your buffs), are based on decreasing the skill check made by the enemy. A Bard's Cutting Words, for example, is excellent!

When a creature that you can see within 60 feet of you makes an Attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature’s roll.

Other options are the Wizard's Portent feature:

You can replace any attack roll, saving throw, or ability check made by you or a creature that you can see with one of these foretelling rolls.

Or even other spells, like Bestow Curse:

Choose one ability score. While cursed, the target has disadvantage on Ability Checks and Saving Throws made with that ability score.

A final suggestion is to provide cover for your buffed ally. If some wall, or barrier is between both, the effect can't be dispelled, as spells require a clear path.

You can't make spell effects immune to dispel magic by RAW... However, you can make them less likely or more difficult to dispel.

Upcasting

By using higher level spell slots it makes dispel magic less likely to counter the spell as the enemy caster would have to make a roll against it to dispel it. Even if a spell does not have an extra benefit from casting it at a higher level you can still do so. It is akin to the Heighten Spell metamagic from 3.X/PF rules if you are familiar. If your allies can find a way to bestow disadvantage on that caster's roll that would increase the likelihood of failure to dispel your buff as well.

Since the caster's roll is a d20 and you are a Divination wizard you can use your Portent to replace their roll.

Counterspell

Have counterspell prepared to cast against the enemy caster's dispel magic.

Nystul's magic aura

With this spell, you can mask the existence of said spell, making it much less likely that your enemies will target it. If you place the spell on yourself or an ally creature it might be an incidental target though. This would only really be an option for permanent effects like major image or teleportation circle but can be used on shorter durations if necessary.

Two ways to peel this onion

giving a one-time or one-round protection from dispel magic.

1. 
   

   For one round of protection, have another caster cast silence on
   the spell caster.

   
   
   

   This does two things.

   
   
   
   
   
   • It makes the enemy caster move
   
   
   • It voids verbal spell components
   
   
   • Your buff stays up for one round, at least, depending on initiative order. Maybe longer depending on what the enemy caster then does. Caveat: As you noted in a comment, if the enemy caster is a sorcerer, and if they have the subtle meta magic, they can get around this.
   
   
   
   


2. 
   

   Stun the caster, or use hold person on the caster (if humanoid).
   
   There are a few in game effects that stun a creature, which will prevent them taking an action; the spells that do this tend to be high level, and the stun lasts a turn for most of them. (This stun thing would be easier with a monk in your party). The reason I suggest using hold person on the caster are twofold.

   
   
   
   
   
   • When held, they can't take an action. Dispel Magic requires an action to cast.
   
   
   • You want the caster taken out quickly.
     
     Your assassin will get advantage, and critical hits, when making a beeline to hit the caster. But even if that first strike doesn't kill the caster, the caster is still unable to take an action to cast dispel magic.
     
     
     
   
   
   
   

   Pre-loading silence or hold person in a ring of spell storing can make this easier to do, since you have a sorcerer with you as well.

   
   

For either of the above to work, you need to identify the caster as such before you put up your first buff. That may not always be possible.

To make your buff dispel-proof ... that's a bit harder, per @Slagmoth's answer.

A few Cheesey methods

1. Cast the Spell inside of a Cube of Force

• Setting 4. Spell Effects cannot pass through the Barrier.

Dispel Magic cannot get in to the spell. This also means that the spell cannot escape, but technically correct is the best correct.

2. A bunch of people casting Anitmagic Field

If you surround the spell effect in people willing to uphold an 8th level spell for however long you want, then it can't be dispelled. I don't know where you're going to get that many bored 15th level characters, but that's your problem.

3. A bunch of people casting Silence

Fine, fine. This isn't "dispel proof", but it goes a long way. Dispel Magic has Verbal and Somatic components. If you can't speak, you can't cast. The problem is that Dispel Magic has a 120ft range, so you're going to need a LOT of casters in order to make enough 20ft spheres to keep your spell effect out of range. Also, Sorcerers can bypass this problem entirely, if they have subtle spell.

Or combine them! Have a 20x20 room made out of Cubes of Force, then you only need one casting of Silence to make the room dispel-discouraged.

Cast a Glyph of Warding in conjunction with Counterspell.

Set the trigger: "Whenever any creature attempts to Dispel X enchantment"

From Glyph of Warding:

...When you inscribe the glyph, choose explosive runes or a spell glyph.

...

Spell Glyph: You can store a prepared spell of 3rd level or lower in the glyph by casting it as part of creating the glyph. The spell must target a single creature or an area. The spell being stored has no immediate effect when cast in this way. When the glyph is triggered, the stored spell is cast. If the spell has a target, it Targets the creature that triggered the glyph...