There are three questions you need to answer when determining if you should use an online solution. How much data do I have? How important is my data? What are the company requirements for keeping the data safe?
The most important thing to consider before deciding to backup online is your internet connection. Data files can be massive and it's essential that your internet connection can provide upload speeds that enable an efficient and reliable backup. Organisations with poor connections will struggle to back up on time, and when data requirements change, can result in poor backup ability. A good internet connection which supports a high upload rate (business-grade at a minimum) is a must for online backup. Without a higher-speed connection the performance of your network may degrade, heavily impacting your users.
Security is often a concern when considering options for online backups. When dealing with an external backup provider, make sure you investigate what security systems they have in place, where the data is actually stored and how reliable their equipment and security policies are. Be sure to deal with a reputable business - this is no time to be cheap. A good choice for a backup provider is one which supports your existing backup application as well as being local to your organisation for faster recovery.
Are your systems mission-critical? If so you should consider Disaster Recovery as a Service (DRaaS). DRaaS vendors are full service providers, which mean they not only store the backup data, they will restore your entire system if you encounter a disaster. This effectively becomes an off-site disaster recovery environment which you can deploy as part of your DR plan.