Q:

Is using Inspiration to give the enemy Disadvantage an issue?

I have been using Inspiration in my campaign for many things, especially though for awarding players for good roleplay, being creative with what 5e offers, and making sessions fun.
All has worked rather well and I found the rules to be adequate until one of my players used a spell which granted a saving throw. With success would come victory, with failure great sorrow. For me this is one of those occasions where spending Inspiration makes perfect sense as the character (in addition to the player) is throwing in all his weight to succeed.
However, reading the RAW in the PHB page 125:

If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. Spending your inspiration gives you advantage on that roll.

When the player announced he wanted to spend his inspiration I came up with the solution of having the enemy roll at disadvantage. This was a spur of the moment thing and I know that as a DM I am well within my right to do this. But if I was to allow it once, I'd want to make it a house rule but not before checking the long-term impact on the game.
Hence I came up with the following reasoning. All of the rolls you get advantage for are made using d20 + Ability Modifier + Proficiency + Misc Bonus if we abstract the miscellaneous bonuses from races and magic items a bit. Seeing as your spell save DC is only 8 + Proficiency + Ability Modifier + Misc Bonus, it seems that you would benefit more from having an enemy roll a save at a disadvantage than roll yourself with advantage. However, I guess it all comes down to what defences your enemy has and I am not keen on drafting an enemy population with a Gaussian distribution regarding their save bonus just to work out the Math.
My question is twofold:

Did I miss anything in the rules or my considerations which could make my argument above void or alter it significantly?
Do you think granting players the ability to impose Disadvantage to one enemy (same rolls as they would get Advantage for) rather than having Advantage themselves is a major issue regarding the game mechanics?

A:

Proceed with Caution, you are entering dangerous territory.
Expanding inspiration in this manner will make it more powerful (by definition). Adding dramatically more options to any ability will do this.
There are also some soft differences between granting success versus granting failure. All party members succeeding once will have a very different feel than a monster who fails four to six consecutive checks.
But these aren't the biggest threat you're going to face. That one belongs to the pure casters...
Save or Suck
While an attack roll on a spell versus a saving throw for its target often seems like an arbitrary distinction, they aren't always. There is a category of spells often referred to as "Save or Suck," which almost universally favor saving throws instead of attack rolls. Spells like Hold Person, Banishment, Feeblemind, Entangle, and Polymorph all inflict near-catastrophic effects on a failed saving throw.
Under your proposed house rule, an optimal tactic would be for players to transfer their inspiration to the full casters in the group (Wizard, Cleric, Druid, Sorcerer), who then uses it to cast repeated save or suck spells on any monster who stands above the crowd.
This has two very negative consequences:

Players are now under social pressure to transfer their inspiration elsewhere, rather than using it to help their own characters shine.

"Boss" fights become incredibly difficult to stage, as any boss has to deal with multiple saves against incapacitating effects with disadvantage applied.

This is not a fatal flaw...
This is not a fatal flaw to the house rule. Monsters written to be boss monsters typically have auto-save abilities, because even without disadvantage save or suck spells are pretty devastating.
In addition, exploiting it requires players to play at a somewhat optimized level. I would not be surprised for an individual group to either not see this tactic, or to be chivalrous about not exploiting it.
But be careful. Make sure that everyone involved is aware that you may back this change out early if it causes problems.

A:

Yes, this is fine.
You seem to have noticed an inherent symmetry in how aggressive spells work — a spell will either:

requires an attack roll to hit

or

requires a saving throw to avoid the effect

But never both.
Since the rules say you can use Inspiration to get advantage on the first kind of spell, it only seems balanced to be able to use Inspiration on the second kind of spell too, right? That's what I see you intuitively figuring in how you made this ruling, as you've described it in the question.
So yes, there's that natural symmetry, and extrapolating from that symmetry to let PCs impose disadvantage on saving throws for spells the PC casts isn't going to unbalance anything. It is thematically consistent too, since spending Inspiration reflects performing better due to being inspired, and making the saving throw attached to your own spell have disadvantage is making your spellcasting perform better, just like getting advantage on a spell with a to-hit roll is making your spellcasting perform better.
I would make one suggestion, though: only allow Inspiration to impose disadvantage in this case, not in every situation where an enemy is making a roll. While it makes sense that Inspiration could be used "on my own spell" without breaking anything, allowing PCs to use Inspiration to impose disadvantage on enemy attack rolls, or enemy saving throws to avoid effects not caused by the PC, would be contrary to Inspirations theme of personal performance, and massively expanding the power of Inspiration. That would likely become unbalancing, and I would not open that Pandora's box.

A:

I would say that giving your opponent disadvantage compares reasonably with giving yourself advantage. (If I'm not mistaken) advantage/disadvantage works out to a +5/-5 bonus/penalty on average. If you're giving yourself a +5 to hit, save, or ability, giving an opponent a -5 to hit, save, or ability would be equally influential.
Having said that, I think the only issue is, should your inspiration be able to influence anyone besides you. In the other cases, your inspiration is causing your character to act better, due to being inspired. A way to work around this, I would say (house ruled, of course) to add a flat +5 to the DC. The only issue here is, is this a multi-target spell? In this case, I would allow it (the increase in DC) to affect only one opponent. He is so inspired, he has been able to focus the spell more strongly on one opponent, making it harder for him to save, and make it a normal save for everyone else.

