I can think of only one beta that convinced me to pick up the full game, and that was for Gotham City Impostors. For the majority of games, I think betas work against them.
Generally, betas don't sell me on a game. It's mostly just an excuse to play a game that I might not buy otherwise. I played a handful of hours in the original Destiny beta, and felt that was enough for me. If I'm unsure about buying a game, I'll read reviews and watch videos of the game before making a decision -- betas are supposed to be tests for the developer's infrastructure, and so as the warnings always clearly state, the game's state at that point in time is not indicative of final quality at launch, so I take it as such.
There haven't been a ton of betas that made me go from not interested to sold, but a few of them have increased my excitement for a game. Recently both the Splatoon 2 and ARMS betas both did just that. The inverse seems to happen more often than not, though, as games like Call of Duty: Black Ops III made my excitement drop considerably after getting a hold of them. Ultimately, it just comes down to the quality of the game. If the game is fun, inventive, and feels fresh then most of the time the beta will excite me. If it's more of the same, or dull, then I'll lose interest and move on.
I don't really partake in beta tests but I did try the Destiny 1 beta. I found it terribly boring so it ended up helping me save my money.