"The cloud" can mean many different things to different people. For the purposes of this answer, I'm speaking of public cloud enterprise solutions like Salesforce.com, AWS, Azure or Microsoft Office 365. There are many business applications that are currently run from public cloud environments, and they are usually more safe and secure than anything you run on premise. Remember, these companies (and the data centres the run from) rely on your trust that their service will perform and be safe and secure, and will back that trust with certain contractual commitments. If anything happened to make you lose that trust, you could switch away from them and they could go out of business.
People often think that the cloud is mostly about cost savings, for example moving from capital expenditure as an up-front cost to lower ongoing operational expenses. However you should also be thinking about other parallel factors like convenience, the ability to scale, the access to new functionality through automatic upgrades, and the ability to work in new ways that these applications will provide. The ability to rapidly prototype, or to run up quick new solutions, is one of the benefits also.
Cloud isn't a magic bullet. You need to think about the business problem you're trying to solve, and working backwards from there. Having said that, most companies these days are thinking cloud-first when it comes to their business and IT operations. If you want a free assessment as to what can work for you, reach out to us at firstname.lastname@example.org