Yes, cryptocurrencies do have value.
To be a medium of transfer of value, a cryptocurrency has to have value. This comes from the cost of acquisition. If a coin is Proof of Work, then a miner must spend money on hardware and operational costs for their mining setup in order to acquire coins. This way, coins are imparted with a value determined by the cost of production. A more rudimentary introduction of value can come from an ICO: if someone pays money for a coin, it is reasonable to think that they will not want to sell it at a loss. Any value which is introduced to a coin beyond these value formations comes from the price that someone is willing to pay to buy the coin. That price is informed by scarcity.
The value of a cryptocurrency changes over time due to the scarcity of the cryptocurrency. Cryptocurrencies are a resource and they are a limited resource. The price of a resource is the cost of acquiring it from the market. This price is affected by the scarcity of the resource. Scarcity is the gulf between a limited supply and a potentially limitless demand. There are criteria of acquisition for any resource. For example, the criteria of acquisition of a normal stone are simply to walk outside and pick one up. These criteria are easy to satisfy. However the criteria of acquisition for gold are far harder to satisfy: to acquire gold you must either set up huge mining operations or pay the price of gold to buy it off the market. Why are the criteria of acquisition for gold so much harder to satisfy than the criteria of acquisition for a normal stone? This is because gold is more scarce than normal stones. The more scarce a resource is, the harder it is to satisfy the criteria of acquisition. This also means that if scarcity is introduced to a resource, the criteria become harder to satisfy. When it comes to the coins of a cryptocurrency, the criteria of acquisition are mining/staking or paying the price of the coin and buying it from an exchange. These are the criteria that will be affected by the scarcity of the cryptocurrency.
There have to be limits on the total supply of a resource for it to have scarcity. This much is pretty obvious: in day to day life, access to the air we need to breathe is easy and the total supply is effectively limitless. No one charges us to breathe air because no one would pay for it. It is vitally important that a value-bearing currency has a limited supply. Fiat currencies achieve this by having the central authority control currency creation and then enforce regulation with a strict legal system. There’s a reason why being caught with a counterfeit US dollar can come with a 20 year jail sentence, even if the person didn’t counterfeit it themselves. Cryptocurrencies similarly need to limit total supply of coins and control coin creation, but they do it through the code itself.
Scarcity can be demand-induced or supply-induced. Demand-induced scarcity occurs when the overall demand for coin is increased, but the total supply remains the same. This could be because the population wanting the coin grows or because the individual demand for the coin grows. Supply-induced scarcity occurs when overall supply is reduced. For example, some cryptocurrencies burn their transaction fees rather than give them to the miners or stakers, which means the total supply of the coin goes down when it is used. To understand how the scarcity of a coin might change over time, we need to have an understanding of what drives demand for the coin. You can read more about how the utility of a cryptocurrency creates demand for the coin in the answer to the question, What key features of a cryptocurrency induce scarcity and hence affect the price of the cryptocurrency?
People want cryptocurrencies because of what they can do with them. A simple example is the ability to transfer value across borders without the need for intermediaries. If someone wants to do this, they will need to buy coins, thereby introducing demand for that coin. The things you can do with a cryptocurrency are the utility of the cryptocurrency. Whenever someone wants to use a coin, they’re creating demand, which in turn induces scarcity for that coin. As we said in the answer to the previous question, scarcity is what informs the price of a cryptocurrency.
We can see how utility creates demand by going through an example. Bitcoin is used as the underlying currency for most of the cryptocurrency world. If you want to buy another cryptocurrency, it’s usually easiest to buy BTC first and then trade the BTC for the other cryptocurrency on an exchange. Let’s say I want a particular altcoin (a contraction of alternative coin; any cryptocurrency which isn’t Bitcoin) and I only have fiat. It is rare to have some way of buying an altcoin directly from fiat. Even if there is, I would most likely have to pay a significant premium on the market price of the coin. The easiest option would be for me to buy BTC, send the BTC to an exchange and buy the altcoin using the altcoin/BTC trading pair on the exchange. I am creating demand for BTC when I buy BTC using fiat. Bitcoin is used to buy other cryptocurrencies because it is the most common reference currency for the whole cryptocurrency market. This utility for Bitcoin creates demand for the currency from me, even if I never want BTC itself and only want altcoins. This is just about the simplest utility for a cryptocurrency. There are many more, much more involved, much more interesting utilities for cryptocurrencies and more are being developed all the time.