A more interesting question would be 'is a borderless world desirable?' And the answer to that question is 'no!'
You can have a borderless world in two ways: The first is that there are no governments. But what is a government? A government is the attempt to uphold certain laws within a certain geographical area. It is the attempt to have a monopoly of physical force within that area. Because to uphold laws requires that you can sanction law breakers and the government does that by using force. To understand the connection between force, government and the law, it helps to think of an example:
If you choose not to pay your taxes for an extended period of time, some government official will send you a letter and probably later come to your house to say that you need to pay your taxes. If you refuse, a policeman will later show up and confiscate your things or throw you in jail. If you refuse to follow, or you block the entry to your house, the policemen will enter your house by using force or drag you away and handcuff you by using force.
This example may seem extreme but it illustrates a very important principle: Government require force to uphold their laws. If there is no credible threat on the use of force, the law is meaningless.
(This doesn't, however, mean that you cannot have rules without government. Because rules can also be implemented by social pressure and exclusion from certain groups. This is what separates government from society: Government uses force to implement its laws. Society uses social non-violent sanctions.)
So: In order to have no government you would have to rely totally on non-violent methods to implement the rules of society. First of all you would need everybody to respect everybody else's property rights. And secondly you would have to agree on a method (i.e. a court system) through which genuine disagreements could be solved. And everybody would have to respect the court's decisions. But this is dangerous because it grants a lot of power to the judges in that court system - however you decide to design it. And because different people have different norms and values, there would often be a difference in opinion between the parties in the case and the judges. History also shows us that it is dangerous to grant this much power to one person or one institution.
If just a single person decides not to follow the orders of the court and the court decides to implement its rules forcefully, this system would cease to be a society and start to be a government.
The second way you could have a borderless world would be to institute a one-world government. So instead of no government, you have one government for the entire world. This brings us back to the problem of too much power in too few hands but it also creates another, huge problem:
The problem with a one-world government is that it would be very hard to see whether the government was making very bad decisions. If a government in today's world makes very bad decision, there is a tendency for people and capital to move abroad. That is why millions of people fled DDR (East Germany) after the Soviet Union occupied it. They were forcing a planned economy down on the Germans and this led to poverty and poor opportunities for families and individuals to design their own life. That many people were fleeing the country was an objective sign that the policies of the DDR were bad policies. If we were living in a borderless world, we would not observe this since there would be no place to flee.
So because it would probably be very undesirable to live in a borderless world - no way to enforce rules (no governments) or no way to tell if rules were bad (one-world government), it is hopefully not going to happen. And I don't think it will, either, because most people in the world still cherish their sovereignty.
And the people saying that if humanity dies out, the world would be borderless, are wrong. Most animals also have territories and laws which are implemented forcefully (e.g. in an anthill or in a group of lions), so the animal kingdom also have borders which are of the same form as we humans do. They also have governments.