According to an interview of Casey Kasem that I read, the entire Mystery Inc. team is wealthy.
Scooby is blue blood old money. His family is pedigrees champion stock. But most of his money came from an eccentric dutchess who left her entire estate in trust to Scooby's family. As such, with all assets tallied, he is the wealthiest.
Norville "Shaggy" Rodgers is also blue blood old money. His family owns several estates, castles, etc. in Europe. They also own vast quantities of land in the US, and own or have controlling interest in many businesses.
Scooby and Shaggy were caricatures of students attending Brown and Princeton Universities at the time the original show premiered.
Velma is old money, but from a more salt of the earth stance. Her family has owned bookstores for generations, again going back to Europe. The specialized in rare and antique books. They did suffer numerous financial setbacks due to various issues, but were always able to persevere and rebuild. This was due to, (and also reinforced), their down to earth nature and work harder and smarter mentality.
Velma represented M.I.T.
Daphne's family is new old money. Their wealth came from railroads and industry in the US.
Daphne represented Yale.
Freddie's family is politics money. Like Daphne's family, they made their money after the US was founded. Compared to the rest of the gang, Freddie's family would be considered new money, as they have only been wealthy for 5-6 generations or so. (Post civil war)
Fred represents Harvard.
Freddie was the "fearless jock" and defacto leader. He did have a knack for traps.
Velma was the tech guru with a penchant for forensics, investigating.
Daphne was the "face" of the gang. Skilled at persuasion and snooping, she still frequently fell into the role of resident "Damsel in distress".
Shaggy was the undercover mech genius. While Freddie may have done most of the driving, Shaggy owned, maintained, and enhanced/upgraded the Mystery Machine. He collaborated with Velma in building their mobile crime computer/lab, and helped Fred with trap construction more than any other member of the gang.
Scooby was of course the lovable goofball mascot, as well as the gang's "bloodhound" for sniffing out clues.
During the original series from the late 60's and early 70's, the whole gang were trust fund kids who didn't need to work. Though they often took on odd jobs as part of their investigations. (That money, if any, most likely supplemented the food expenses.)
[During the late 70's and on, their previous experiences helped mold their growth into individuals, and led them to their own chosen professions with the dissolution of Mystery Inc. until they were reunited and chose to revive their detective agency.]<---this is my personal opinion and in no way derived from the interview I read. I also chose not to go into the darker side of the gang's alleged partaking of recreational pharmaceuticals.