Can a country be held responsible for a crime committed by a private company? Let's say a private security company in Syria like Blackwater https://en.wikipedia.org/wiki/Blackwater_(company) committed a war crime, can the government of Syria be held responsible? What are the criteria by which a country can be held responsible for a crime committed by a private organization?
"Can a country be held responsible"...
Let me stop you there. Countries are sovereign. They can do whatever they want, and the only ways to get a country do something are 1) to ask nicely and hope. or 2) go to war.
If you haven't invaded and occupied a country, you can't force the people in that country to do anything.
Now with respect to war crimes. Justice is principally in the hands of the winners. When you say "be held responsible" I ask "responsible to whom?" Now suppose a Blackwater mercenary commits some war crimes while acting on behalf of the US government. Can Syria put the US president on trial? Well, yes, if Syria can invade the USA and arrest the president. Is that going to happen? No not in a thousand years.
In international law (which as I emphasise is just a collection of agreements between sovereign states and not like a system of law in the usual sense). There is an international criminal court, and countries can agree to use it. But if a country chooses not to send their president to the ICC, there is nothing the court can do. It has no powers of arrest.
However, crimes against humanity and crimes against peace are the responsibility of everyone. If a leader commits a crime against peace by the mechanism of a private company, that person is just as guilty as if they had used the regular army.
Yes, states can be held responsible for crimes committed by private companies. Imagine if it had been differently. A rag-tag dictator could just pay a private company to do all the dirty work for him! In your example, if the Syrian government enlisted Blackwater to, for example, ethnically cleanse the country of some minorities, then Syria would be liable for the actions of Blackwater. That wouldn't absolve Blackwater or its minions of culpability; they would also be culpable. Note that it wouldn't matter that Blackwater is (or was) an American company as it would be acting on behalf of the Syrian state.
From Britannica's chapter on the responsibility of states (emphasis mine):
The rights accorded to states under international law imply responsibilities. States are liable for breaches of their obligations, provided that the breach is attributable to the state itself. A state is responsible for direct violations of international law—e.g., the breach of a treaty or the violation of another state’s territory. A state also is liable for breaches committed by its internal institutions, however they are defined by its domestic law; by entities and persons exercising governmental authority; and by persons acting under the direction or control of the state.
These responsibilities exist even if the organ or entity exceeded its authority. Further, the state is internationally responsible for the private activities of persons to the extent that they are subsequently adopted by the state.
A common example is massacres. A state's armed forces may have regulations prohibiting massacres of civilians. That would not absolve the state of responsibility in case an army unit "exceeds its authority" and commits a massacre.
The International Law Commission in 2001 adopted a set of draft articles spelling out exactly what a state's responsibilities for "internationally wrongful acts are". I'll cite the articles which are relevant to your question:
Article 1 Every internationally wrongful act of a State entails the international responsibility of that State.
Article 2 There is an internationally wrongful act of a State when conduct consisting of an action or omission: (a) is attributable to the State under international law; and (b) constitutes a breach of an international obligation of the State.
Article 5 The conduct of a person or entity which is not an organ of the State under article 4 but which is empowered by the law of that State to exercise elements of the governmental authority shall be considered an act of the State under international law, provided the person or entity is acting in that capacity in the particular instance.
Article 8 The conduct of a person or group of persons shall be considered an act of a State under international law if the person or group of persons is in fact exercising elements of the governmental authority ...
Article 5 and/or 8 would be applicable in the Blackwater example.