Updated based on The Amazing Spider-Man 2
No. The spiders would not have successfully given powers to just anyone.
In a video log, Richard Parker explains that his own DNA had been fused with that of the spiders. Only someone with shared DNA (such as Peter, or likely Ben Parker) would successfully gain powers from the spiders.
Original answer based on The Amazing Spider-Man
In-Movie answer: No, not just anyone would probably gain the powers.
Dr. Connors mentioned that every cross-species DNA merge resulted in death. Even with what should have been a stable formula, Connors body did not react to the formula the same way - changing him into a giant Lizard while Peter simply gained powers.
Not-Quite-In-Movie answer: There is probably something special about Peter.
There is a fantastic write-up about The Untold Story which was cut out of the movie. It discusses hints of deleted scenes that were in trailers and pieces them together with the final film to theorize the supposed "Untold Story".
Months ago I told you guys that I heard rumors that The Amazing Spider-Man would be making a simple, but huge, change to Spider-Man’s origin. No longer would the spider bite change Peter Parker into a superhero. Rather, the spider bite would activate something already within him that would make him a superhero.
The first major hint is still in the movie. Curt Connors is talking about how every other subject upon whom cross-species DNA merging was attempted died. He does not know that he is speaking to the one success story. But how did Peter survive? The movie leaves this sort of dangling there, but the clues are in front of your face. Peter was bitten by a spider... a spider that Peter’s father bred. A spider like the one under glass in the film’s prologue. A spider like the one on the chalk board in his father’s office.
In the first and last trailers we hear a man - clearly Mr. Ratha - whispering ‘Do you think what happened to you, Peter, was an accident? Do you have any idea what you really are?’ That certainly sounds like a reference to Peter’s genetic destiny, and a clip that I don’t believe is in the finished film. Judging by the whisper I’m going to guess - and this is just a guess - that it is Ratha’s dying words to Peter after the Lizard does him in. This is a movie that seems like it should have at least one info dump dying declaration in it, and this would have been it.