They went out of business, because
- They were sued
- They were banned from ghostbusting
- No paranormal activity until the events of Ghostbusters II
- People didn't believe them
From the script by Harold Ramis and Dan Aykroyd (1989.02.27):
- They were sued
Speaking to Hardemeyer
VENKMAN: No, that's what I want to talk to the mayor about. We did a little job for the city a while back and we ended up getting sued, screwed and tattooed by deskworms like you.
- Banned from ghostbusting
Speaking to Venkman
STANTZ: Hey, hey, hey, stresshound! Are you nuts? If anybody found out about this we'd be in serious trouble. The judge couldn't have been clearer - no ghostbusting.
- No paranormal activity
Cars and trucks swerve and hit their brakes as Dana runs into the intersection and snatches up the baby. She hugs it close, deeply relieved, then looks at the buggy with the dawning awareness that the supernatural has re-entered her life.
At that time, Vigo hadn't taken control of Janosz yet. This may be somewhat weak evidence, but there was obviously not enough paranormal activity for the Ghostbusters to be needed.
- No believers
Winston and Stan "play" Ghostbusters on kids' parties, with moderate success (they were hoping for He-Man instead):
BOY: My dad says you're full of crap.
STANTZ (stopped cold): Well, a lot of people have trouble believing in the paranormal.
BOY: No, he just says you're full of crap and that's why you went out of business.
And to Venkman, who wanted to speak to the mayor
HARDEMEYER (bristling): Look, you stay away from the mayor. Next fall, barring a disaster, he's going to be elected governor of this state and the last thing we need is for him to be associated with two-bit frauds and publicity hounds like you and your friends. You read me?
If the public, the voters, believed the Ghostbusters, the mayor would rather want their support. Instead, he wants to have nothing to do with them.