Clarke spoke out on the subject in February after being subjected to bigoted abuse at Ibrox over his Catholic background while in charge of Kilmarnock.
Clarke praised the “very strong” reaction of Rangers chairman Dave King at the time but believes a number of clubs could do more to fight the issue.
Rangers will play Legia Warsaw in the Europa League on Thursday in front of at least 3,000 empty seats after UEFA punished them for “sectarian chants” from supporters in a qualifier against St Joseph’s last month.
Speaking on his own experiences, Clarke told BBC Radio Scotland: “Rangers reacted very well to it. They are trying their best as I’m sure the people from Celtic are.
“Everyone tries their best but sometimes maybe the little Uefa ban will make some people think about what they are going to say in the future.”
When asked whether the Scottish football authorities should do more, Clarke added: “I think the clubs have to clamp down on it. If the clubs could do it themselves, it would be great.
“And the clubs do try. But sometimes they maybe have to try a little bit harder.
“I thought the move by Ann Budge at Hearts last year to close that little section of the ground was a brave move and I think it’s worked well for them because it’s given people a little lesson. You can’t go into a football stadium and behave like an idiot.
“You have to be supportive of your own team rather than trying to be negative towards the opposition.
“Just go and support your team, enjoy the game and hopefully have a good day out, that’s what football should be.”
Ayrshire-born Clarke – who spent almost 30 years in England before returning to Kilmarnock in 2017 – hopes football and society in Scotland will move on with the help of people speaking out about sectarianism.
“That night at Ibrox, I lost it to be honest,” the former Chelsea player said. “I was really upset. And I was upset. I think anyone who sees the video can see I was genuinely upset, because I had come back to Scotland and times had moved on.
“Rangers obviously were much more inclusive. I did have a conversation with them about maybe going there. It was not about getting the job, it was ‘would you be interested?’ And I was because they are a big club, a great club.
“And then to get that on that particular night…
“The chants started fine, I was whatever I was, no problem. It was their night. We got hammered and it was the first time we had had a spanking from Rangers since I came back.
“So they deserved their moment and I was happy to take that.
“But the sectarian singing, you don’t need. If it happens in the street it’s a crime. If it happens in a football stadium with so many people doing it, then suddenly it’s not a crime. That can’t be correct.
“It’s something I feel strongly about, I spoke about it at the time and I would love to think that in the future in years to come it won’t be an issue in Scotland.
“And hopefully, with education and time, we can get better and it will become a thing of the past.”