As you know by now, the cloud has everything to do with the internet. Without an internet connection, you will not have access to your resources. The question is, "how mucehh internet connection do you need?". Depending on the service, the numbers will vary as will the costs. Let’s look at the services by category (Web Apps, Platforms, and Infrastructure) as well as consumption method (Desktop and Mobile)

Web Apps

When accessing Software as a Service, it’s generally in the form of an application that you’re accessing via a browser. In some cases data transfer of the web application to your local browser will be no more than that of a normal website. In other cases, the application may be utilizing some of the features of the HTML5 standard that require downloading media or storing data. Having said that, one of the benefits of HTML5 are the lower file sizes of images displayed as SVGs vs. the standard PNG images. 
Because of the variety of web applications and methods for developing a web application, it’s impossible to set one static number on the bandwidth requirements. 
A Hosted Exchange interaction (comparable to Gmail, Yahoo Mail, or any other Email web application) requires a minimum 4.2 kb/sec per user. 
Applications like SharePoint Online, which could be compared to a Dropbox solution, have a higher minimum at about 8 kb/sec per user. 
Communication Applications generally top the requirements because of the constant streaming and uploading of audio or visual content. Most voice protocols require 80 kb/sec per user and video protocols come in at as much as 350 kb/sec per user.

In all cases, these numbers should be doubled to account for a daily peak of twice the average of what expected to be required and even then these numbers can be considered bare low experience minimums. Often in the case of Communication Applications, the bandwidth usage is almost nothing when there is no active usage (having an interaction).

With Web Apps, as with normal websites, if the bandwidth is available there will be no throttling (unless it’s from your ISP) and your browser will attempt to retrieve the data as fast as possible. The limits then become how fast your browser can process the download. Real bandwidth usage in your environment may spike to the full size of the site download if there is no throttling and you have plenty of bandwidth. The usage of caching will lower your actual bandwidth usage if you’re visiting an application where most of the elements do not change.

Your bandwidth costs in accessing Web Apps are no more complicated then the cost of service from your ISP.

Platforms

Platform as a Service such as Virtual Servers can be accessed in a variety of ways but the bandwidth requirements are more predictable. In the case of a Microsoft Azure, Amazon EC2, or comparable remote server offering, the bandwidth requirements will depend on the type of server which will dictate how you interface with it.

If you are accessing a Window server, you will likely need an RDP connection to access the desktop environment. The minimum bandwidth required for a usable RDP session is 26.4Kbps per user or session. If you are accessing your server over a VPN connection, you may need as much as 512Kbps per site.

In the case of a Linux based server, you may be able to connect and interact with that server with a SSH connection. That connection would require as little as 346bps per session to maintain a connection.  You may need to connect to a desktop environment on your Linux server and in that case would be using a VNC connection. According to the makers of VNC, 33Kbps is sufficient for accessing a remote desktop with low graphics requirements.

The numbers noted are all minimums and performance may not be sufficient for a desktop quality experience.

Besides your ISP costs, your bandwidth costs in accessing Platform services can be slightly complicated. In some cases you will be paying for data transfer from your server to the rest of the internet (which includes your remote connection to the server). Many provider counts both upload and download bandwidth usage, some only count download usage.

 Infrastructure
 

Cloud Storage is one of a few solutions categorized as Infrastructure as a Service. There are no real minimums or recommendations for cloud storage. Your experience with the back and forth data transfer will based on whatever available bandwidth you have. The more bandwidth you have, the faster your data can be transferred. 

Just as with PaaS above, you will likely be paying for the bandwidth of your data transfer to the cloud provider. The difference here is that in many cases you do not pay to upload the data, but pay more to download it. 

Desktop vs. Mobile

The significant difference between a desktop device vs. a mobile device in accessing "the Cloud" are the limits of the mobile network. This can become less significant when you are stationary and connected with to a Wi-Fi network with high speed internet access. With some devices, this may be your only option as not all devices support the 3G and 4G networks necessary to get an internet connection back to your cloud resources. 

With real speeds reported at between 1Mbs and 3Mbs can provide more then significant pipe to support your usage to a number of services.

Costs when you access the cloud via a mobile device may compile if you are using a carrier that charges for your increased bandwidth usage.