Cloud storage works with every kind of internet connection. Whether you have WiFi or LAN internet, you'll be able to carry out an online backup onto a cloud storage provider without any problems. If you're planning to back up large video files, for example, a quicker internet connection is better. Lots of big image files will also take longer to upload, regardless of internet connection, but of course a slow connection will take longer. If you're also using the cloud storage for multiple devices at the same time, a reliable and fast internet connection will mean your cloud storage will work better.
In short, cloud storage works with all internet connections, but your usage will determine the internet connection speed you'll require.