Christos Matskas has an article on securing Azure blobs and containers:
All communication with the Azure Storage via connection strings and BLOB URLs enforce the use of HTTPS, which provides Encryption in Transit. You can enforce the use of “Always HTTPS” by setting the connection string like this: “DefaultEndpointsProtocol=https;AccountName=myblob1…” or in SAS signatures, as in the example below:
https://myblob1.blob.core.windows.net/?sv=2015-04-05&ss=bfqt&srt=sco&sp=rwdlacup&se=2016-09-22T02:21:41Z&st=2016-09-21T18:21:41Z&spr=https&sig=hxInpKBYAxvwdI9kbBglbrgcl1EJjHqDRTF2lVGeSUU%3D
To protect data at rest, the service provides an option to encrypt the data as they are stored in the account. There’s no additional cost associated with encrypting the data at rest and it’s a good idea to switch it on as soon as the account is created. There is a one-click setting at the Storage Account level to enable it, and the encryption is applied on both new and existing storage accounts. The data is encrypted with AES 256 cipher and it’s now generally available to all Azure regions and Azure clouds (public, government etc)
There’s some good information here, making it worth the read.