What is the value of the connection between different technologies? Between people?
That's asking about the value of something intangible. It's a bit like asking how much Coca Cola's brand recognition is worth or trying to quantify the value of writing. In some cases it's easier to quantify an exact value because someone can purchase distribution rights, etc. associated with that intangible good. In this case, it really isn't something that can be bought and sold, at least not in its totality.
If you wanted a better answer, instead of something abstract like "it's priceless", the best I could give you is to total up the value of all of the organizations that work toward realizing or enabling that idea. We all pay some amount of money to an ISP for this connectivity or we purchase hardware (modems, ethernet cables, wireless routers, etc.) that enable its use, and those are all quantifiable. That's what people are willing to pay for the connectivity so it at least gives us a good lower bound.
Another possible way to approach an answer is to look at wealth generated as a consequence of that idea. On one hand we could calculate the value of industrialization by adding up the cost of all the factories, machines, etc., but you could make an argument that the value lies not just in the cost alone, but what has been enabled as a result. All of the profit that someone can derive as a result of those connections existing is some potential value that could have been charged by the people selling the machines, etc. You can get yourself into a bit of trouble with this because you can argue that agriculture is worth just about all of human productivity because almost none of modern society would be possible without it.
However, that's a far more difficult problem because it becomes difficult to truly separate out all of those different numbers and invites a lot of argument over whether some of them should be included or not. Many companies are the result of several intangible ideas being realized and it's going to be difficult to attribute a percentage each is responsible for. But you could look at companies like Facebook that can only exist in large part due to this connection and include their worth (or some part of it) as a part of the calculation as well.
One can also look at the value of what newer technologies replaced as well, particularly if they're less expensive than existing solutions. For example, most people book flights online through a website. Previously this was done over the phone or through a travel agent. While those things haven't gone away entirely, they've largely been replaced just as computers largely replaced switchboard operators, which at one time required a lot of human labor.
I'm not going to sit down and try to come up with an exact dollar amount, but I do think that adequately explains how to go about quantifying the value of the Internet Protocol in the way that you've defined it. The companies that build the hardware necessary to support that connection are collectively worth some hundreds of billions of dollars, if not well into the trillions alone. Throw everything else it enables on top of that or other jobs that can only exist because these connections are possible, and I would think it's well into the tens of trillions of dollars.
If this question is in relation to BitCoin I don't think it's a very good argument to attack the idea of the worth of BitCoin or any other cryptocurrency. First, even though it is a digital currency, the design is such that for any coin (or fraction of one) that exists, it has exactly one owner. Since these can be exchanged, the value is whatever the current owners of those coins can sell them for, same as any physical coins. You can calculate the overall value by looking at the current going rate of exchange on the market and multiplying that by the number of coins that exist, similar to how the value of a company is derived from the sale price of stock. Obviously that's not perfect since trying to sell all of a company's stock or all of a currencies coins at one time is going to adjust the value, but it's not just as simple as adding up the book value of all of the company's physical assets either.
I suppose it's still a simple question, but I don't think it's one that has a simple answer.