Let's assume that bitcoin was taxed in form of a percent-based tax on every mined bitcoin. How could government enforce this tax, considering that the whole idea of bitcoin transactions is to remain anonymous? If it was implemented, what would be the economical consequences for bitcoin miners? When all countries worldwide would implement this taxation on bitcoins, would people stop using it?
It is unlikely that a tax on bitcoin mining will ever occur, because it's so hard to enforce. Phillip proposed the most viable solution for enforcing such a tax, but it breaks down the moment that bitcoins are transferred outside of the country of origin.
Since it is theoretically impossible to prove who generated a bitcoin and bitcoins can always be transferred to any part of the world any policy that is not enforced by every part of the world, ie being built into the bitcoin policy itself, is unlikely that any country could enforce a policy. So long as any one country does not enforce a taxation policy that one country can be used to 'launder' bitcoins.
As a side note I say that bitcoins are supposedly anonymous because I suspect most bitcoin generation is not all that anonymous. If one is not actively obscuring it using something like TOR an IP address can be associated to a region or even, with assistance form low level service providers, a specific individuals home. One could argue that in theory this would allow the goverment to directly tax generators of new bitcoins by associating their IP address with the person who purchased the network connection associated with the IP. In reality though this is not realistic. There are many situations where this system would be imperfect, such as when many individuals sit behind a router and are all associated with a single public IP address. More importantly there would be a non-trivial expense and privacy outrage in trying to track someone by their IP and in the end if a goverment did try this approach Most miners would be financially encouraged to hid their IP behind a TOR.
A more likely method for 'taxing' bitcoins would be when they are exchanged for actual goods. When someone exchanges a bitcoin for a dollar, or a pizza, or some other commodity etc. Any time something is purchased with a bitcoin within a country a tax can be charged for that transaction. This is easier since most transactions can be associated with a physical location, either the location shipping a good or the home state where a company is incorporated in etc... in theory. There are still some complications.
First is that a tax only works if you know who to tax and it's not difficult to trace them. If bob agrees to pay x bitcoins (or fraction of a bitcoin) to his friend Fred down the street to buy Freds old bicycle this is not going to be easy to trace. The goverment will likely write off this as not worth taxing, either making it untaxed, or making it technically taxed but doing little if anything to prevent these sort of under-the-table deals. Instead the goverment will focus on taxing larger operations that exchange bitcoins. That is to say their tax will apply when bitcoins are touched by Amazon, or Paypal, or commodity exchange companies etc. for bitcoin transactions. These big name cooperation, which are easy to associate with an IP and watch the transactions of, will be responsible for the majority of bitcoin transactions and as such getting tax only on these easily traced corporation is acceptable even if under-the-table deals still happen on occasion.
The second, bigger, issue is a question of how to create taxes that don't force companies out of your country. Imagine the (foolish) policy of the US goverment saying that any bitcoin transactions processed by Paypal which happened in the us (which is to say the transaction was sent to an IP address who's serever was physically located within the US) was taxed. The obvious response by Paypal would be to move all their servers outside of the US and not worry about the tax. If instead the goverment attempted to force Paypal to pay a tax on any bitcoin transaction it processes, regardless of where the server was physically located, Paypal would complain since it may be handling a transaction between two individuals in Japan that the US doesn't have a right to tax.
So instead the law would likely say that they expect a tax to be payed by these large company whenever they process a transaction for a US citizen (actual law would be a bit more complicated since someone living in Germany who buying a service from another German citizen who just happens to also be a US Citizen shouldn't be taxed, but the general idea of calling out which transactions the US claims tax on still applies...)
The goverment would then have to leave it up to the companies in question to determine who was a US citizen and rather taxes apply. Now in theory this means someone could still used a TOR to hide their identity to do a transaction. However, the majority of people may not know how to do this or be too honest to do so. Ultimately since most money transfers are between large corporations a large amount can still be taxed. Again the goverment may not be able to prevent someone from hiding their identity, but it will be treated as any type of tax fraud if it happens.
There is a related issue of some countries setting up tax havens. Some tiny country X will accept US citizens bitcoins and exchange them for dollars without paying US taxes. The goverment will have to work to limit these sort of tax havens, but again since most transactions happen between large companies, that have to abide by the local government's laws to operate within the country, or by honest citizens who either are too moral, or too afraid of repercussions, to try to break the law the majority of bitcoin transactions should still be taxed.
So as to your original question, of what would be the effect...nothing too interesting. Obviously the interest in bitcoin will decrease if it's heavily taxed, and nations that tax bitcoin transactions heavily will see less use of bitcoins. There would be some increased effort into trying to avoid tax havens and prevent 'fraud' due to the easy of it happening and the difficulty of proving it, but i suspect for the most part the goverment will be willing to write off a certain degree of such fraud as it will be cheaper to allow some of it to happen then to try to prevent it.
Since every country will have a different law for how things commodities are taxed there will be all kinds of odd situations like US companies keeping their bitcoins 'overseas', or doing everything they can to justify a transaction as being taxed by a country with less (or no) tax on bitcoins then having the transaction count as being taxable by a country with a higher bitcoin tax rate, but again companies already do this sort of tax law loophole and tax haven logistics with regular taxes.
Honestly the most interesting part to me is the inherent volatility of bitcoin. An attempt to charge a tax on a bitcoin transaction using your countries preferred currency (dollars, yen, rubies, gold pieces, bottlecaps) will be made harder by the fact that there could be a non-trivial difference in price between morning and evening, but again countries have seen this sort of thing with penny stocks and know how to handle the fluctuation.
On a political level one or two big countries taxing bitcoin could, theoretically, drop the bitcoin's value by a disproportionate amount. As the original users of bitcoins were my fellow geeks interested in anonymous cryptocurrency, who may take offense to goverment regulation of bitcoin, and those wishing/needing anonymity, including criminals, who would not like the increased regulation of bitcoin which would hamper that anonymity. There is a chance that this could drive current bitcoin users to start using one of the many other cryptocurrancies out there over bitcoin. Which in turn would likely lead to countries rewriting their 'bitcoin tax' to be a more general 'cryptocurrancy tax' which tries to cover all forms of cryptocurrancy out there. The debate over rather a particular cryptocurrancy is covered by a countries tax law could be interesting.
Finally since I personally believe bitcoin is still a bit of a bubble if a large country were to enact a bitcoin tax in the immediate future there is a small chance that could put enough pressure on bitcoin to 'pop' the bubble and see it rapidly drop, but that's mere speculation on my part and a topic better covered by the economics page then politics :)
Implementing a mining tax
When the Bitcoin developers are on board with the tax (or can be forced to comply), they could add this into the Bitcoin protocol. Whenever a bitcoin is mined, x% of that bitcoin are immediately transferred to the wallet of the government. Which government is determined by the IP address of the miner.
When the Bitcoin developers refuse to cooperate, governments could mandate to do this manually by law. All bitcoin transactions are public, so everyone can see which wallets got bitcoins through mining and which of them already paid the taxes on them to a government wallet. That means it is possible to see which miners comply with the taxation law and which don't.
Using a non-taxed bitcoin in commerce would be considered tax fraud. However, bitcoins are fungible. When one has 2 BTC (one taxed and one not) and transfers 1 BTC to a different wallet, the protocol doesn't say which one got transferred. The solution is to simply transfer the tax debts together with the bitcoins. When a wallet has a balance of 10 BTC, an outstanding tax debt of 1 BTC, and transfers 2 BTC to a different wallet, that tax debt is divided proportionally. The first wallet is still expected to pay 0.8 BTC tax and the receiving wallet now has 0.2 BTC outstanding taxes.
Many wallet softwares will likely add a feature which warns a user when they receive a transfer from a wallet with tax debt and ask the user if they want to pay the tax debt immediately. This is in the interest of the user, because until they pay the debt their bitcoins will have less value because they can't be spend legally until the tax burden is paid.
Now you could argue "but bitcoins are anonymous. How would the government find out I practice commerce with untaxed bitcoins"? Well, if you want to buy something tangible (buy groceries, pay rent, convert into fiat money at an official exchange...), you must at some point make business with an officially registered company. Those can and will be audited by government tax investigators. They won't accept money from untaxed wallets, because doing so would get them into a lot of trouble.
Effect on the value of mined bitcoins
Let's assume that the government won't just hold on to the bitcoins and actually spend them quickly. In that case the number of bitcoins in circulation wouldn't change and so wouldn't their value. It might go down slightly, because miners can no longer drive up prices by hoarding 100% of the bitcoins they mine.
However, there is now less incentive to mine. The value of a bitcoin minus tax could fall below the energy cost required for mining one. But that point is anticipated anyway. An intentional design feature of the Bitcoin protocol is that coins get harder and harder to mine.
So a bitcoin mining tax would make the point at which nobody will bother to mine bitcoins anymore happen slightly sooner.
If you ignore all the technical details, then you figure out that trading in bitcoins is a zero sum game. When you pay money for a bitcoin, all you do is hope that someone else will be willing to pay the same or a higher amount of money to you. The total of all money given to sellers of bitcoin is exactly equal to the total of all money paid by buyers.
But then we need to subtract cost: The cost of mining, the cost of trading. So all in all, money is lost. If bitcoin is taxed in any form, then more money is lost. The buyer now must hope that someone is willing to pay the same or higher amount, plus extra money for all the cost involved. So a tax, no matter how it works, would make trading in bitcoin less attractive.