Let's say you wanted to make a useful character that never makes attack rolls (or at least not attack rolls for damage). How would you do it? How close to doing it do you think you could come?

The Bard class waves "Hello".

Technically, you could use cantrips that require saving throws rather than attack rolls to attack. Probably not what you’re looking for, but I thought I’d mention it.

There's a firbolg cleric on critical role that pretty much never attacks anything. Like Geann said, support is basically what the bards are made of (especially Lore bards). Any spellcaster who focuses on buffing/debuffing spells also work for this. Even Mastermind Rogues with their double help action can be auseful supporter even though they waste their Sneak Attack.  

You can use any spellcasting class -- as long as you only choose spells that require a saving throw rather than a spell attack roll. There are plenty of those to choose from.

be a lore bard with the tandem tactician feat

here: https://www.dndbeyond.com/profile/Level20WarlockMaster/characters/36911318

College of Glamour Bard (LGT character)

Action: Blindness/Deafness, Faerie Fire, Bestow Curse, Vicious Mockery, or even just Help.
Bonus Action: Mantle of Inspiration

I was annoying enough that our DM would focus me even though I wasn't the one dealing (much) damage.

To not have any weapons, and for unarmed strikes to not deal damage wouldn't you just need a low strength stat?

but we want the character to be usful, but not really dealling damage.

Okay, I'll attempt to make one

The support Character There's like one attack and so many uses

imma delete now, sorry to anybody who didn't see it

I am doing something nearly exactly what you describe with my Lore Bard.  He wears leather armor and carries a rapier, but he only pokes something with his rapier about once every three rounds.  He can't stay up on the line because he has too low an AC and not super high HP.  His main contribution to the party is to cast Bane and Cutting Words.  Sometimes he casts Dissonant Whispers or Counterspell, but he generally doesn't run up front unless he sees an opportunity to "off" a creature with his rapier and get the final blow.  This frees up another character to engage a new critter.  But just as quickly, he steps back and does his usual Bard Thing.

He can also support our Rogue just by standing there next to whatever the Rogue is fighting.  He just has to watch out that the critter might decide this crazy jibbering lightly armored dude is tastier than this pokey-pokey-sticky-sticky rogue.

Quote from MusicScout >>

I am doing something nearly exactly what you describe with my Lore Bard.  He wears leather armor and carries a rapier, but he only pokes something with his rapier about once every three rounds.  He can't stay up on the line because he has too low an AC and not super high HP.  His main contribution to the party is to cast Bane and Cutting Words.  Sometimes he casts Dissonant Whispers or Counterspell, but he generally doesn't run up front unless he sees an opportunity to "off" a creature with his rapier and get the final blow.  This frees up another character to engage a new critter.  But just as quickly, he steps back and does his usual Bard Thing.

He can also support our Rogue just by standing there next to whatever the Rogue is fighting.  He just has to watch out that the critter might decide this crazy jibbering lightly armored dude is tastier than this pokey-pokey-sticky-sticky rogue.

 Vicious Mockery.  Also, 3 levels of mastermind on this guy would give that tasty bonus action Help, which may or may not help out the rogue. Plenty of possibilities there. 

To the OP, Clerics are another class that can provide great support without ever making an attack roll. If you want to get even more explicit and say no damaging, Bard and Cleric can both excel in those roles. 

Rogues can do quite a bit as well, since most of their damage comes from sneak attack.  A rogue can even do a fair job tanking with dodge and the right build. It's not something that I would recommend doing on a regular basis, but it's possible.