There is never any reason, and I mean any reason to release a game into the public in its alpha build. Alpha testing should be done in-house and in-house only. When a game is in its alpha state it's usually around the time where the devs are going to meetings and discussing what is and isn't working in the game. This amounts to things like certain gameplay mechanics, UI layout, maps and other portions that don't belong or need to be added to game. It would be a bad idea to release this for public testing as many people may get the wrong idea about the game's quality due it being a largely unfinished product. But don't let that stop you from making a kick-ass trailer.
Now, in the beta version, the game is practically complete. Whether or not this warrants a public beta depends on the game. For many genres such as FPS, TPS, and action-adventure, there is no reason for an open public beta. A much more sensible approach might be to have focus-group testing, where a number of gamers are allowed to play portions of the game at the studio to give feedback that way. However, there are games where it might be a good idea to do a public beta and those are:
1: Competitive RTS games. Starcraft is the most predominate example of this. Starcraft 2: Wings of Liberty received a limited public beta test and SC2: Heart of the Swarm is currently undergoing this right now. This is because these games are played on such an incredibly competitive level, that Blizzard needs to make absolutely sure that the multiplayer is as balanced as they can possibly make it before release. They need feedback from pro-players and higher ranking amateurs alike. Sure they'll probably still make small adjustments as the meta-game evolves, but right now they have to do the best they can.
2: MMOs. Not all do this but some take this route. Firefall is currently doing it, for example. I don't know all that much about MMOs, but I imagine reasons may include balance issues, in-universe trading and travel, and server capabilities. I can see how this might be a good idea, as there's the risk of people getting the game, finding ways it's broken and becoming pissed off and quitting, which, with MMOs, could financially ruin a company.
3: Possibly a game that is using a brand-new technology. I know that Quake 3 did this and I even beta-tested it myself back in 1999. This was probably to make sure that the new engine is running smoothly on a variety of machines. Also, the test was not the full game, but rather a small collection of maps that id Software felt best represented what the Quake 3 engine had to offer. Also sometimes a studio will allow a beta test for publicity. Recently some game journalists were invited to play the first few hours of Bioshock Infinite. Spoiler alert: They said it was awesome.
Also I believe that no one should ever have to pay for a beta version.
