What are the supports that don't become completely useless when behind?

janna

knock ups and shields are useful no matter how ahead or behind you are

thresh

his lantern hard counters stupid teammates

Janna is probably most notable. She is pretty static in her ability as a support. She doesn't snowball like the rest of her team when things are good, and she, similarly, doesn't fall that hard when she is down.

Braum: He can still apply stacks to people

Lulu: Tons of buffs and debuffs on enemies and allies alike.

Sona: Healing and a very strong poke.

Thresh: A lantern to rescue people.

Most traditional supports can say relevant as long as they are not extremely far behind. The CC/shields/heals/ally-buffs/enemy-debuffs built into their kits will remain useful all points in the game, with or without a lot of gold and items. Non-traditional supports that rely on stats that are heavily gold dependent like damage/HP/MR/AR or items for CC/ally-buffs/enemy-debuffs because they them built into their kits will have a harder time and typically fall behind more easily.