"I did not take the initiative to tell my coworkers"
In life you have to make choices on which information you are telling people, you simply don't tell everything about your life to every coworker. And if something cannot reach the ears of some of your coworkers then it cannot reach the ears of any coworker at all.
The fact that you've said matters, the fact that they've asked doesn't. At least as far as feelings go. But I'm pretty sure you wouldn't be prosecuted for perjury if you simply lied or said something ambiguous like "well, I have a side gig that pays more than the weekend's pizza".
People resent colleagues that are being paid a higher salary for similar roles. If you told a coworker that you earn 10% more than he does, you could be sure he would resent either the company or you, unless you are undisputedly a more important asset to the company or a more senior colleague, which you are not, because you've only been there for three months, and writing wasn't your job before.
Having a (relevant) secondary source of income is something unchartered in general. But I've worked with people who already had generous government pensions and indeed needed no salaries, and I was sometimes annoyed that if the company would go bankrupt I would be desperately looking for a job while these guys wouldn't even flinch. Every time I needed these guys to go an extra mile and they didn't, I would remember their pensions. A similar thing may start happening to you, so start presenting yourself as a hardworking person at this job.
Other answers claim that "You basically told your co-workers that you're laughing at them", I don't agree with this and would not infer it from what you've written, but take that just as a piece of evidence of how much people can have strong feelings about other people's wealth. Think that people made a petition asking for Jeff Bezos not to be allowed to return to Earth.
"I also told them that I joined this company only because I want to do something related to my passion (which is writing), and not because I need a salary, as the income generates from the online business is greater than my salary."
Do you think you didn't come across as thinking yourself better for having an online business? Well, I'd think I would be a much better version of myself if I had a second gig that made more money than my 9 to 5 job, and probably your colleagues would think the same. You've threaded very uncarefully there. Not an apology-worthy problem so far, but it does make it difficult to earn someone's sympathy.
"How should I handle this and protect my reputation?"
Why do you want to protect your reputation?
Are people actually spreading lies about you? If so, start taking notes (save a file with details, and timestamps) of any encounter you have that you suggest or prove you are being defamed.
Are people just saying you are an asshole? Then Don't be an Asshole!. I stand with the thesis that you've earned some ill will towards yourself, but this is somewhat fixable.
You've just arrived at their company, people have barely started to get to know you. They are at the phase when they've done the welcoming reception (everybody in a small company should try having a small chat with every newcomer) and now it's regular life. So people aren't just acting nice to get to know you anymore, as you are not news anymore.
In this short period though, having a lot of gossiping is also not good, no matter the reason, as it's pretty much a sign of a toxic workplace. Every now and then, consider if you'd be doing what you love better in a place you'd love more.
That being said, zero workplace gossip is something unheard of for me. But in general, people should give more importance to how you've treated them and how you act around them than what they've heard about you. I've once started dating a person who told me that at first she really didn't like me, long story short, she had a friend who actually had a previous grudge against me (and I never got along with that person).
Having touched the universal "Don't be an Asshole!" piece of advice, think for a bit if people have some more reason to have a bad opinion about you. Are you nice to everyone? Have you tried being friendly? Have you been finding yourself in arguments too often? Have you complained about the food in the places people invited you for lunch? Are you acting like you know stuff better than everyone else (doesn't matter if you have 20 years of experience in a workplace of teenagers, you don't get to act like the authoritative voice until you have proven yourself to these people).
The financial point gives people a reason to "want to dislike you", not a reason o "dislike you", so the bad news is that a lot of people are looking for these reasons. If instead of giving people what they are looking for you actually prove yourself a nice person and a good worker, things can change a lot in a few more months.
Regarding the money thing, do consider doing some charity work or donations. Whatever does not look ostensive, does not look like trying to show off but does allow you to show off. Maybe tell everyone in the company that there is a charity you contribute to and give them the directions to do the same should they so desire. Seems like a simple trick to earn some sympathy to compensate for the earlier mishappening.