NPCs won't attack you without some kind of provocation.
Causes for NPCs to attack you would be (there might be more):
- You ran or bumped into someone hard enough for them to retaliate if you don't walk away fast enough
- Wearing a bandana (it's sometimes easy to forget to remove it and not notice). People tend to get suspicious and will ask you to remove the bandana. If you ignore repeated comments regarding the banda, you might get attacked.
- You wander into a place where you're not wanted (someone's camp, for example). They'll often warn you to leave and then attack when you don't.
- Opposing gang members (O'Driscolls/Lemoyne Raiders/etc.)
- A scripted scenario
Most of the time, simply running away will limit any impact on your honour scale/bounty. While you could waste time beating them as gently as possible or trying to hog-tie them, this increases the chance that law enforcement officers will turn up and make the situation worse.
Given enough looting of dead enemies, you should have enough funds to comfortably pay off the bounty should you be affected.
The game doesn't really punish you for earning bounties - you just pay them off. However, having a higher honour level does enable some missions later on in the game.
To prevent the "wanted" status, you need to stop the witness from contacting the law. Do this by either killing them or convincing them to stay quiet (an early mission takes you through this scenario). If your fight is in a town, then all you can do is run and get out of the red area as quickly as possible, and then pay the bounty if you're identified as being guilty.
You have good friends. Even if you are without money on missions that require it you will find prices are lower than they would normally be, there's a cheaper version of the necessary item, you can choose an alternate course of action, or friends will give you cash.
Minor spoilers below for three Chapter 2 missions, but nothing too far into the game's storyline.
For instance, one of the earliest missions in the game A Quiet Time culminates in you running from the police while thoroughly inebriated. If you are caught you won't have to pay the $10 bounty because your buddy pays it off for you.
Two missions later you're invited to shop for horses in Exit Pursued by a Bruised ego. You are obligated to sell off a horse and purchase a new one with the money. There are three options available: A Morgan horse for $15, an American Standardbred for $150, or a Dutch Warmblood for $450. If you are (unsurprisingly) unable to afford the $450 horse, and the moderately expensive $150 horse. You may notice that the sale price of the draft horse you sold ($5) will not be enough to allow you to purchase the cheap $15 Morgan.
If you are so insolvent you cannot supply the other $10 to afford the $15 horse you will have to reload the save and instead choose to stable the draft horse rather than sell it.
Lastly, in one of the mission The Sheep and The Goats you will need to purchase a Rolling Block Rifle - an expensive weapon typically selling for $189! Thankfully your buddy John Marston will call in a favor and have the gunsmith sell it for free, thereby saving your wallet and ensuring you can purchase the firearm.
There are a variety of ways to continue a mission even if you don't have the requisite cash. In most cases it will not require reloading a mission's checkpoint, but it is not impossible to force that hand either. No matter what, the game will make accommodations for Arthur Morgan.
Best Answer
Your only options for clearing a bounty are either paying it off, or surrender and go to jail. The latter is only possible when your wanted level is low, so if you've been shooting up some law enforcement you will likely be shot dead if you surrender. To clarify: there is no decay or decrease in bounty over time!
When you surrender, you'll have to wait out your sentence in jail. This page claims that you could be rescued by friends whilst your in jail, but I haven't seen that happen myself. Granted, I haven't spent a lot of time in jail.
To surrender (from IGN Wiki):