Green Day are considered to be a punk rock, pop punk or alternative rock group. Although I find that most bands can change the genre that they "belong to" from album to album, and sometimes even song to song.
"Wake me when September ends" doesn't exactly fit the same genre as "Brainstew" or "Basketcase". But they are all great songs.
Personally, I'd class them as a punk rock band - although, as is the case with most artists these days, their sound is more accurately described as a blend of different genres.
Are Green Day "real" punks?
I first started listening to Green Day when I was in high school, and although we had a few punks at the school I went to, they seemed to be more into punk bands like Rancid, Bouncing Souls or NOFX (some also listened to "old school" punk like The Ramones and Bad Religion).
In their eyes - Green Day had sort of lost face because they had become really popular, and started experimenting with different styles (think the track "Misery" on the album Warning).
And I guess that's one issue that has caused musicians in any genre a bit of a problem: As soon as a band gets recognition, airplay, and a large following - they become too mainstream to be associated with the grassroots.
This has happened to a lot of punk bands over the years (The Offspring for example). In their underground years, no-one would have disputed their punk credentials.
Nowadays, bands like that are labelled "pop-punk" and are looked down upon.
I don't think this is very fair. Especially because, when you think about it - what could be less "punk" than deciding who does and doesn't fit into a scene or genre?
Why I think Green Day are a punk rock band
No-one can dispute that Green Day had an impact on the punk genre - so, for that reason alone, I think they deserve to be recognised as a punk band.
I also think they should be congratulated for being brave enough to absorb different sounds and influences in their music. Whilst I don't personally like anything that they recorded after Dookie, it would have been worse if they got lazy and simply released album after album that sounded exactly the same!