Depends on where you are sitting. If you are in the Poltronissima (and particularly Gold) the patrons will generally be better dressed although there are always a lot of tourists and it is summer so I wouldn't say that is a constant. If you go back to the gradinata (stone steps), it is much more casual with people picnicking, etc.

Personally, I wear trousers, polo shirt, sport jacket, loafers without socks and I am comfortable both temperature and style wise (and I generally sit in Poltronissima Gold).