Multiple people have brought up the idea that you run the risk of brushing up against laws regarding tax evasion, and money laundering. It's important to make sure you avoid these risks. Here are some ideas:
Regarding tax evasion, realize that the gift is not tax deductible, even if you were to run the payment through a non-profit intermediary such as the aforementioned "The Give Initiative", or even your own charity that you might start. If you tried to deduct the gift (personally or as a business expense) you would enter into tax evasion territory. An exception would be if you hired the recipient and paid him as a contractor or employee, but then of course he would have to pay income tax on the amount received. That being said, fortunately there aren't any tax concerns from the recipient's point of view for gifts, since in countries such as Australia (and the US as well as many others) the recipient is not taxed on gifts received, anonymously or otherwise. (And this is why it's illegal to give gifts to employees or contractors without considering them income.)
Regarding money laundering, laws are centered around detecting money obtained illegally being converted into "clean" money. (Cleaned means the original illegal source has been changed to look like a legal source.) Typically money laundering involves cash since once you enter the funds into the banking system there is a record of it in your name. Transferring money from bank to bank in a traceable way (check or wire) is a pretty good way of announcing you aren't trying to launder money, since it is easily traceable back to you, and you can be audited to confirm where the money came from. This does not necessarily change if you use a single intermediary, as long as government authorities can still easily trace the flow of money if they ever wish to. In other words, it's fine for the recipient to not know the original source of the funds, but it must be the case that if the recipient were questioned, he could reveal he received the funds from the intermediary, and the intermediary could reveal the funds were received from you. (Unless the authorities are allowed to inspect the banking transactions themselves, which is likely in some countries.)
Given that, the suggestions would be:
- Do not deduct the gift (or consider it a business expense, etc).
- Do not transfer cash.
- Realize that the source of the funds should be easily traced back to you (by authorities with proper access, not the recipient).
- Select a trustworthy intermediary, perhaps an attorney as suggested in StephenG's answer.
- Make sure the intermediary has enough information to convince the recipient this isn't a scam, and that the money will be returned to you if it isn't accepted.