I agree with some others that Rapiers aren't so much the issue as Dex is. I've house-ruled several things that limit or exchange Dex:
Initiative can be made using Dex, Int, or Wis, player's choice. The idea being Dex lets you react quicker, Int gives you knowledge of tactics, and Wis lets you read the situation and opponent better. You can't use more than one unless your archetype lets you use Int or Wis, in which case you can also add a second. With multiclassing and the proper archetypes, you could have all three (even Cha also with Swashbuckler...), but then you are pretty much just sculpting your character to go first.
Any armor that lists Disadvantage on stealth limits other skills such as acrobatics as well. Medium armors limit the Dex mod you can apply to checks just like it limits AC bonus. Medium Armor Master feat removes these.
There are enough other was to redefine how your AC is calculated and ways to buff AC that Dex is useful, but hardly the god-stat it once was.
Many other abilities Dex assists with can be accomplished in other ways, even if not so subtle.
But to address the OP:
1) removing the rapier as a weapon - but I don't like this at all
2) renaming the rapier as "broadsword" as in AiME, just for the flavor of a medieval setting, but it doesn't solve the problem of Dex being unbalanced as a Stat
3) removing the "finesse" quality from the rapier, and giving it the "light" quality, so as to allow two weapon fighting with a rapier and a dagger or a shortsword
4) toning down the damage die of the rapier from d8 to d6
5) modifying the official rule, and stating that you use Strength to determine the bonus to damage even with finesse weapons: if you want to use Dexterity to determine the bonus damage with finesse weapons, maybe you have to select a specific, homebrew Feat (maybe called Agile Fighter?) that allows you to do so.
What do you think?
Are Dex and rapier/finesse weapons a bug or a problem in your games and campaigns? If so, how do you handle it?
1) not necessary really to remove it completely (unless it doesn't fit your game, such as in my world)
2) right, doesn't solve your problem with rapiers
3) only limits finesse melee weapons to d6 only.
4) as with 3 above, limits finesse weapons to d6 maximum. I think rapiers were chosen as
one weapon that would do a d8 to match other martial weapons.
5) this wouldn't bother me as a player, but you could house-rule it to finesse weapons add half Dex modifier (round up or down, your preference...). So with a Dex 18 you have the +4 to attack rolls, but only +2 to damage. Your homebrew feat could remove the limit and restore it to full Dex modifier to damage maybe.
For my game, rapiers don't fit the feel of play as in our history they didn't exist until the 1500s or so. I play more in the dark and middle ages type settings, so I just don't have them in the game.
Hope you work something out.