Are there guidelines for punishments for crimes?

Yes, within Explorer's Guide to Wildemount

Explorer's Guide to Wildemount, released in 2020, has several lists of crimes and punishments associated with its various kingdoms and empires.

For example, amongst others, multiple places have laws against trespassing, murder, and "ownership of sentient humanoids". Here's the punishment for these three crimes within the Dwendalian Empire:

• Ownership of another intelligent humanoid creature: 12–20 months incarceration and a fine of 1,800 gp.
• Trespassing: 12 days incarceration and/or a fine of 215 gp.
• Murder: 8–15 years incarceration and/or a fine of 6,000 gp, or execution.

The same list of crimes for the rival kingdom (the Kryn Dynasty) shows:

• Ownership of another intelligent humanoid creature: 10–20 months incarceration and a fine of 1,500 gp.
• Trespassing: 5 days incarceration and a fine of 120 gp.
• Murder: 5–10 years incarceration and/or a fine of 4,000 gp, or execution.

The book has a somewhat extensive list of crimes (about 12 per country) and punishments for the major political players in the setting (three major countries). These tables can be found in chapter 2, "Factions and Societies".

For the world of Faerun, at least, this will probably vary from place to place. Waterdeep, the city where the Dragon Heist module is set, does have its own Code Legal that list various crimes and their consequent punishments and/ or fines. Examples:

• Assaulting or impersonating a Lord: death
• Arson: death or hard labor up to 1 year, with fines and/or damages covering the cost of repairs plus 2,000gp
• Assaulting a citizen: imprisonment up to a tenday, flogging, and damages up to 1,000 gp

In a comment to another answer, you write that this is a neutral character edging towards evil, and that he is partying with good characters.

In that case, leave the punishment to his party members. Worst case, they leave him behind while going adventuring without him. Leaving him behind is not what you (the players) want to happen, but it can be the threat that can make him do what the others want.

Other than that, the easiest punishment would be paying restitution. Steal something, pay back threefold. Hurt somebody, pay weregild. Again, his party members should force him to pay rather than just skipping town.

The DMG does not provide any specific guidelines on crime and punishment, however it does suggest that player characters might serve time as part of downtime activity. Check out the downtime activity for "Carousing" for more detail on that.

In addition to the great answers already provided, I wanted to chime in as I feel it is important to note that punitive measures taken against player characters are often interpreted as punitive measures taken against the player, so whatever punishment you mete out needs to be fair and needs to not take the player out of the game. In your question, you state that "[you] can see him causing problems later on through the game (which [you are] fine with)," which says to me you aren't interested in punishing the player.

That being said, while being thrown in the stocks or pillory might be a fine in-character punishment, it also makes that player's character unavailable for play while the rest of the party is out gallivanting. To deal with this issue, I suggest one of two things:

1. Allow him to serve time in the stocks or pillory, but as part of downtime activity. Being that he is likely a hero, or at least associated with heroes (or maybe not... I don't know your game), he might be able to get away with something like the following:

   PC has committed a crime and been caught in the act...
   Magistrate: As punishment for your crime, you are hereby sentenced to three days in the stocks!
   PC: But, m'lord! I have a very urgent and time sensitive matter to deal with regarding a dragon in distress and an evil damsel! I must heed the call of heroism at once!
   Magistrate: Very well, you have a week to complete your task and upon your return to the city you will be expected to serve your sentence.

   • Upon his return, then, he serves his punishment as part of downtime activity while the other PCs are getting drunk at the bar.

2. Make the punishment another quest or reparations to the victim. For example, if he knocked out an innocent man, his repayment might be to recover the man's lost macguffin from the angry dwarf who held a grudge for a slight against his character. Or maybe his repayment is to clean the man's stables (downtime activity, yay!).

Both of these options are, in my opinion, much better than simply saying, "okay, you're serving time in the stocks now. Go do something else while the rest of us slay the misunderstood and mostly not evil lizardfolk." They offer more opportunities for roleplaying and prevent the player from feeling like he is being punished for something his character did. You don't want to go down that road, because it will only lead to problems at your table.

In a lot of medieval legal systems, I think that offense would have been a fine, which he'd be compelled to pay on pain of death. "Blood money" was pretty common.

First, there are no lists of punishments in the rules.

Second, the default D&D 5e setting is a fantasy medieval one, as such, imprisonment is an inappropriate punishment. Imprisonment as a punishment is a late modern development from the early 19th century:

The original purpose of confining a person within a prison was not to punish them, but was a means of keeping the perpetrator of a crime detained until the actual punishment could be carried out. This was usually in the form of corporal punishment that was intended to cause the guilty person pain, such as being beaten with a whip, or capital punishment which used a variety of methods to claim the lives of condemned individuals.

www.crimemuseum.org

A more appropriate punishment might be the pillory or stocks both of which were used to restrain the criminal in a painful position (Exhaustion might be an appropriated D&D mechanical outcome) and to allow the populace to humiliate them by laughing, hurling insults, rotten vegetables, offal and, of course, human or animal excrement.

Here is an interesting list of historical punishments for both criminals and children from around the world. I give thanks every day I was not an Aztec child.

This presuposes that the rule of law is suficiently established that a common assault is a matter for local law enforcement. In medieval times (and some places today) retribution was a private matter. For example, gathering a bunch of friends together and beating the snot out of the perpetrator.

Rules wise, there is no list of punishments for crimes. Each village, city, and hovel has its own law, and the DM is the one who decides how the world reacts to the players. So, it falls to the DM to decide what the law of the land is, and how people will react to being assaulted on the street. In a lawless land, there is no justice. In a big city, I think jail time. This can lead to an interesting challenge for the party to break their friend out of jail.

Now, if there was a contextual reason why this player chose to attack an innocent man, or if you guys decided as a group that doing whatever you want is fair game, you can proceed with an in game response that you feel is appropriate to the city.

However, if this player is just being disruptive for the sake of it, and his actions fly in the face of what you guys determined you wanted out of the game, then a better response would be to talk to him about why he is being so disruptive and how you can make the game more interesting. Then, let him off the hook with a slap on the wrist.

You say he is going off the rails, which leads me to believe it is the latter. Always remember that you describe what happens in the world. If your player is being disruptive, it's your right as the DM to say "Why don't we take a break for a quick second before I resolve that," and then take a second to talk to him in private about what you both expect from the game.

This is left almost entirely to the DM. There is a section in the DMG p114 that broad strokes a legal issue as an Urban encounter but a punishment will depend on the region, ethos and politics. In your given situation depending on where it occurred and to whom it could be as simple as the other peasants don't deal with him friendly anymore, costs go up for him or he is outright denied access to the victim's friends' establishments. It could go even to the opposite extreme and the "innocent" happens to be the son of the governor and the penalty could be death.

The range is limitless and involves thought into who runs the settlement, the motives of said ruler as well as the motives and power (if any) of the victim to persuade action against the PC.

Although I personally don't like FR there were some mentions of a smattering of the laws and some punishments in previous edition supplements. I completely ignored the FR specific region details of the Sword Coast Adventurer's Guide but you might have some luck in there.