I have not experience with taxis in Prague, but I just flown there the previous weekend and can at least tell this:
- Everybody agrees that Prague Taxi drivers are mostly dishonest.
- Inside the airport, just in front of the exit doors, there are stands of one of the official taxi company (FixTaxi, if I remember correctly)
- The public transport from and back the airport is really good and cheap, you just have to change at a point to take the metro to the city, but it's a minor issue
- This is the company that the airport itself recomended: http://www.prague-airport-transfers.org/, and they have fixed prices.
So, that said, I'd point out that someone has to wait: as obvious as it can sound the plane will land when it will land and your dad will be outside the airport the moment he will be outside; what if you book a taxi, the airplane land exactly on time, and then your father is stopped half an hour for borders checking? It's a reality that can always happen to everyone.
He will have to wait for a taxi or a bus, or the taxi will wait for him. And the latter will be really expensive...
It would be better if you explained a bit more the problem you have -"I fear that he may has to wait for a while", is your dad waitophobic? Maybe he doesn't speak english? Where will you be while he will land in Prague?- but anyway, a few options:
Public transport, take the 119 to the metro station and then the metro to the city. The metro station is a the end of the bus ride, the stop is named Nádraží Veleslavín, and when he will step down from the bus he will be in front of the metro entrance. Ticket machines are outside the airport, just in front of the exit (impossible to miss) and the bus stops in front of them, they have the choice for many languages and are really easy to operate anyway. And if your father has a contactless credit cards, it's just a matter to push one button and put the card in front of the machine. Same for metro ticket.
Taxi and he doesn't speak english: I'd go for Prague Airport Transfers. Just ask your father to call you when he will be gone past border checking.
Taxi and he speaks English: you still can do the same as above, or he can just book a taxi from the stands I mentioned before. I wouldn't fear him have to wait for long; taxis go where the money are, and the moment an airplane lands is money landing; I bet that there will be a few taxis outside the airport waiting for a passenger booking at the stand.