I have a slightly different approach.
Adventurers are treated differently than most folk. First the characters should be gaining a reputation if they are doing anything unusual.
Second Adventurers while respected (they have a lot of power compared to most people) they are also feared (they have a lot of power compared to most people) and are thought to be a little crazy (They rush in or get involved in crap others ignore).
So a constables reaction is push it up the chain of command because adventurers get involved in stuff that will get us killed.
I prefer to play the constables mostly honest (yes! there is corruption but not blatant in most towns) and hard working, they help but consider adventuring stuff beyond them and a sure way to get killed. Doesn't matter if this is true that is how they see it. Imagine what fantasy knights go do questing, average soldier has no interest in going to fight dragons, rescue princesses from devils etc. That is for crazy folk.
while a lot of ideas presented here are good they make for a more adversarial role with the constables, authorities. No reason to go that route unless you have too. I have run long urban adventures (they work best when you have only 2-3 players, more fail states to work with) and tend toward the society/ community appreciating the adventurers overall as they Save the day. I also prefer more heroic good guy games, not goody too shoes but real heroes flaws and all.
The characters are sort of like firemen. People are glad to see them come when there is a fire, and sort of glad to see them go after cause this means they are safe and in the end they are not the ones who have to rush into the fire.
There are different ways o define "honest". Take the secret peace from the gentlemen bastards series of books or the fact that sharn's biggest mob boss type is also on the city's ruling council. The guards know where the rest of their pay comes from & the underworld knows why they pay a percentage of their gains to maintain arrangements.. Both sides know that the local police equivalent going to war with the entire underworld would be terrible. Most importantly is that as long as nobody on either side of things is crossing the line things remain peaceful. If someone on either side crosses the line however it's up to that side to enforce the unspoken rules on their own brethren to keep the peace.
The guards don't just look the other way when a criminal element gets away with criminal activity because of a power disparity... They look the other way because the particular criminal activity was within the lines of acceptable & even if it were something they could track down to punish doing so would lead to far more chaos and bloodshed.
An unusual edge case like the family of a guardsman being targeted by the criminal underworld simply would not happen and on the off chance it did someone could politely inform the right people who would then insure that the person who screwed up learned a lesson and the family was compensated with replacement or better... Sure Bob's granny knows her jewels were stolen because she even saw the burglar dash out the window with a rope still tied to her dressing room furniture after she came to investigate the noise... "but granny your eyes were probably playing tricks on you & the rope was leftover from that repair work that was going on. They put your jewelry box away in the safe & are here to finish remodeling your dressing room so lets just put this whole misunderstanding behind us."
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