The best adventure games like Crackdown are InFamous, Prototype and Saints Row IV. Below are 7, each with the reason it is here — and, at the end, what filtering to adventure costs you.
Filtering to adventure means the place and the story are what you want back, not the challenge. It is the right filter when the thing you remember is where you were rather than what you did.
What you get: A world with a reason to wander it, and writing that survives being read rather than skipped.
What made Crackdown worth playing in the first place is covered on the full list. This page is the adventure part of that answer, and only the adventure part.
The other great orb-collecting superpowered city, and it landed a year later.
Wall-running and vertical traversal with no restraint at all.
Super-jump collectathon in an open city — the closest thing to Crackdown's orb loop.
Turns traversal into the whole game by letting you fall in any direction.
The best-feeling city traversal ever made, if you want it polished.
Grapple and wingsuit movement in a huge open sandbox.
More of the same city if that is genuinely what you want.
Mechanical depth. Several of the best adventure games have combat that is deliberately unsatisfying, because they want your attention somewhere else.
Specifically, these are strong answers for Crackdown that are not adventure games, so they are not on this page:
The full list is on games like Crackdown — 9 picks, no genre filter.
InFamous. The other great orb-collecting superpowered city, and it landed a year later.
No. Crackdown released on the Xbox 360, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Anywhere the story and the place matter more than the skill ceiling — point-and-click, walking sims, narrative thrillers and most open worlds. The label alone tells you very little, which is why each pick here says what it actually is.
They split hard. Open-world adventures run to a hundred hours; the narrative ones are often finishable in an evening, and the short ones are usually the better written.
By hand. Each pick was written for the full Crackdown list first, on what the game actually does, and this page is the part of that list that is adventure — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the adventure guide for where to start in the genre generally.