So Spider Man shows up a solves a petty crime. He starts doing this once a week. Although he avoids law enforcement (since he is technically a vigilante, which is illegal), he otherwise doesn't try to hide himself. Many people get pictures of him (including Peter Parker), and publish them. How long will it take until people accept that Spider Man is real, if ever?
Since you clarified in the comments that you're asking about our universe, it's going to take something a lot bigger than petty crime for people to believe.
Spiderman is a well-established part of popular culture. So well-established, in fact, that jokes have been made about other superheroes pretending to be him as kids. At this point, Spiderman is such a well-known character that anyone claiming to be him would probably be treated like someone claiming to be Jesus.
If all he's doing is stopping petty crimes (muggings, carjacking, robbing convenience stores, etc), nobody is going to pay attention. At absolute best, some videos pop up on YouTube: "Guy Dressed as Spiderman stops Robbery".
Now, if something big came up where he was not only seen by a lot of people (like in Amazing Spiderman 2 when he fought Electro in the middle of Times Square), but more importantly seen using his powers, then people would start to believe. One video is a fluke. A few pictures can be photoshopped. A hundred videos of the same event from different angles? Now that's proof. But until then, not a chance.
It would honestly be easier for him to be believed if he took a whole new identity and started calling himself something else, simply because "Spiderman" is so ingrained into our culture.
Pretty soon (within 6 months).
Security camera recordings.
Dozens of eye witnesses, photos and cellphone videos.
And most importantly, the gooey web which some criminals would have on them, would be sent for DNA analysis and it would reveal a human genome mixed with a spider's.
Very soon. Comic books have been preparing people for years. Millions of people want to believe Spider-Man is real. While the first reaction would be one of disbelief, once they see the pictures and hear the stories, they will eventually come to believe it's real because deep down they want to. People believe in Bigfoot and aliens with even less evidence, so why not Spider-Man?