This is tricky because it's about who has the right to be wrong about what.
I think gambling is generally a bad thing. It can be fun, but it can also be generally destructive. While I'm not sure its helpful to throw around words like "addiction", it's pretty clear that many people who gamble are behaving very strongly against their own interests.
Should gambling therefore be illegal? Absolutely not. The problems are threefold - you are stopping the harmless entertainment as well as the self-destructive behaviour; you are raising your (and my) judgement as to what is good for someone else over their own judgement*, and you are introducing the plagues of prohibition, including a criminal class and a corrupt enforcement bureaucracy.
However, despite these very strong arguments, the governments of the USA and many of its States have banned gambling (with various indefensible and illogical exceptions for State lotteries, etc).
One of my more eccentric beliefs is in National Sovereignty. If a foreign state (however constituted) wants to get stuff wrong, then unless it directly affects me, it's really none of my business. They're entitled to do stuff differently; that's what being foreign is all about.
That proposition doesn't flow easily from any theoretical statement of morality or justice. You could build up to it from a concept of democratic rights, but as I don't restrict sovereign rights to democratic states, that doesn't help me. For me, sovereignty is a pragmatic rule, a compromise which reduces the amount of conflict between countries - and conflict between countries is one of the major causes of human suffering and poverty. As such, the principle can be overridden in very extreme cases - such as the Rwandan genocide - but those familiar with this blog will be aware that I am very much more cautious than most regarding "humanitarian violence".
Of course, since no-one is suggesting starting a war to protect the human rights of Americans to play online video poker, I've gone off on a slight tangent here. Mind, we did once fight a war for the human rights of the Chinese to take opium, but even those of us who favour drug liberalisation generally give less than wholehearted approval to that project.
There is a kind of consistency to my views: just as Beryl should be free to damage herself by buying lottery tickets (but I would prefer her not to), the USA should be free to damage itself by prohibiting gambling (but I would prefer it not to).**
Now we come to the tricky stuff. What if an American flies to Britain, walks into a bookmaker's shop in Luton, and puts a bet on a horse.
Well, that's OK, I think obviously. The US government might choose to deal with the visitor when he gets home (but in fact, according to current law, wouldn't).
What if the horse race doesn't run until the visitor has gone home. Can the bookmaker pay the visitor's winnings, by sending him a cheque or crediting his bank account? The question is whether the bookmaker is simply settling a debt (and the fact that the transaction which gave rise to the debt would have been illegal if it had taken place in the US is beside the point, because the transaction didn't take place in the US), or whether the payment itself is a transaction with someone in the US which is in breach of US law.
I think the US government is entitled to consider it the latter. Gambling is, after all, not much other than an exchange of money; if you send a cheque to America in settlement of a gambling transaction, you are gambling with someone in America.
Since you are outside US jurisdiction, you are safe, since the US ought to respect your country's sovereignty.
But if you later travel to the US, their government can justly claim that you have been dealing with the US in a way that is against US law.
To take a parallel but less morally confusing example, if a Nigerian scams me out of a stack of money by claiming to to be MIRIAM ABACHA, and then later comes to Britain on unrelated business, he should be arrested. Exactly what country he was in when he conned me, and what the law is in that country, is beside the point.
When we come to the actual cases that are in the news, most recently Peter Dicks, another question arises. Was he knowingly dealing with the US? I think that matters: if, as far as he knew, he was simply carrying on a legal business, and unknown to him, some of his users were actually in a jurisdiction where the business was not legal, then he hasn't done anything wrong - it is like my very first example of a bookmaker completing a transaction in Luton with an American visitor.
On the other hand, if he is knowingly transacting business with people in America, he is like the second example of the bookmaker sending a cheque to America - the transaction is taking place between two countries and is illegal in one of them. I would think that in the concrete cases existing, this is the case.
The structure of the internet makes it possible to not know the location or nationality of your customers. This makes the question really difficult. I suppose the US government is still entitled to make its own rules about how careful those who come within its reach should be to avoid acting, while abroad, in a way that it considers illegal. But if it does act against those who as far as they know are behaving totally legally within the jurisdictions they are working in, it is stepping over a line of what is generally considered reasonable behaviour of a state. Note it has not yet done that over the gambling question, as far as I can see.
What I'm really arguing against here is the idea that the internet changes the rules - that if what the server is doing is legal in the place where it happens to be sitting, then no other government should be able to do anything about it. It would be nice if it did, but I say that only because I am generally in favour of freedom, and that would bring more freedom. I can't defend it in terms of logic or history, though. The internet isn't the first mechanism to allow people in different countries to deal with each other, and governments have always held that they can restrict or prohibit such dealings according to their own policies.
*It is OK to make a judgement about someone else's interests - as I have done. It is another matter to deny that person their own (bad) judgement
**That is an analogy - I do not claim that states and individuals should always be looked at in the same way.