Ahhh, "How to 'balance' casters vs. non-casters?" The 64,000 gp question.
There are a few threads to pull with this...
#1. As in all things the "B"-word, what do you mean when you say "balance?" Define the kind of "Balance" you mean.
Do your players mean DPR? I think most acknowledge 5e's actually fairly carefully balanced -in the sense of everyone being reasonably even/fair- across all classes for dpr.
Is it flavor? You're never going to change that. One set of classes using magic -which does things that just can't happen in real life [for the most part]. By definition, from "Magic Missile" and "Detect Magic" on up, these classes are "reality bending"...because, "Hi, Magic." One set of [very very small in 5e] classes does not use magic/spells. These two sets will never, anywhere, in any way, "feel" comparable. One group is making balls of fire burst into existence in their hands, causing wounds to close and bones to knit, *POOFing* enemies to other dimensions[!]. One set is swinging their swords and picking their locks.
Other than a certain iteration's creative direction of going full-on "comic book superhero/physically impossible" for stuff for the non-magic users, you're not even going to close that flavor gap...but there will still be a gap. One group of classes can do, eventually, literally anything. One group is bound by [more or less] the physical laws of the material world (so even that half-genie guy with the fire for hair can't just leap across a 30' chasm under their own muscle ability). You -or anyone- are not going to be able to make those two groups "feel" the same...other than, as you note, in a game that simply doesn't include any magic/spell use...or, I suppose, conversely you could also close it by simply making EVERYone a magic-user (which seems the path in the dark forest that 5e decided to go most of the way down).
So, first step, what's this "Balance" your players are concerned over? What does it look like to them? What does it look like to you/the DM? Are those compatible or competing definitions?
#2. Once you know that, the question becomes, "What will be fun, as DM, for you to implement?"
I think you've covered your options, there. Choice A: "Rein in" the casters (which hardly seems fair to the caster players). or Choice B: "Beef up" the non-casters' options/character choices (which is probably the powergamey kind of stuff that admitted 4e-nostalgic non-caster players are looking for).
I might offer an option C: a Combination of the two that doesn't substantially alter much of either, but brings them closer. This, naturally is the least attractive option as it presents the most work for you [the DM] and doesn't really result in either side being fully happy. But that's what compromise is.
2A: Reining in the Casters: Something I've toyed with in the generation of my own game system is -as I believe others have mentioned- extending powerful spells (for me/my tastes, I'd say over 3rd level) casting times.
The simplest I've come up with, though I can't speak to its efficacy in actual play, is adding 1 round of casting time for each spell level above 3rd. So a 4th level spell requires 2 full rounds to cast (taking effect at the end of the second round). 5th level spell needs 3 rounds, and so on.
My goal wasn't so much to deter uber-powered casters, but to promote the Group/cooperative elements of play. "Keep the ogres off me. I'll open us a portal to escape." But would certainly assist your goal if any significant (4th or 5th+ level) spell requires multiple rounds to cast...and has the bonus carry on effect of giving the non-casters something meaningful [or so it should feel, one would hope] to do while the casters are bending reality.
2B: Beefing up the Non-casters: the usual go-to answer to this query..."I'll give the fighters more STUFF they can do!" From BECM on, more levels for Fighters meant more "things" they could do...that were usually very situational and next to worthless. But they were in there! More levels for Thieves meant more skills/abilities and higher success rates of everything.
Then, of course, there were the other "out of combat" elements of the earlier editions of the game, including forming strongholds (or "hideouts" for thieves) and attracting followers/trainees of your class. The latter weren't exclusive to non-casters, though they certainly had more benefit (and I think larger numbers of low-level classed followers) than the casters.
What this all ends up boiling down (or snowballing up) to is "adding complexity"...to the non-caster classes and the overall functioning of the game itself. That's not inherently, of course, a bad thing. Some people like complexity. Some people want as much complexity [i.e. "player choices"] for their non-caster as the casters receive.
It is, however, completely unrealistic. Casters have, literally, dozens of choices (spells to choose, spells to change/prep each day, etc...) at practically every level of their experience and very day of play. Non-casters will simply not be able to compete with that...without using 2A to virtually nullify casters to as few options as the non-casters get. Then, yes, everyone's "balanced" in terms of fairness. Everyone has "the same." That's fair/"balanced," right? But it just makes casters completely void of their flavor.
I do think, and have done in my system generation, giving a few more options [class features] to non-casters doesn't hurt anybody. I throw in a little more "front-loaded" at lower levels than casters get. I throw in features at levels where the casters don't receive any (because they're still getting their spells). And a few more interspersed through higher levels when caster classes simply don't get any (because they're continuing to pile up their magic arsenal).
All in all, the numbers of "features" casters to noncasters are not the same. For individual leveled features, the noncasters seem to have a solid bunch more. But when you look at the number of spells that can be cast in a day, the casters still far outpace non's for possible meaningful action beyond "swing sword/use thieves' tools."
There really is, I don't think, no satisfying way to "balance" noncasters with casters and maintain the feeling of each: either the magic-users become completely useless/one-trick ponies ("nerfed" I believe is the term kids are using these days) or the non-magic types become immersion-breaking super-powered mutants.
Careful application of a bit of A and a bit of B is your best bet, I'd say.
I know that is probably...unsatisfying, but I don't really know what else to tell you.
Hearing what you decide/how you go about it..and the results will be appreciated. Good luck.