Save the key-value string pairs (connection strings and other secrets) in the Azure website configuration information. This sensitive information never appears in plain text in the web.config file.
http://azure.microsoft.com/blog/2013/07/17/windows-azure-web-sites-how-application-strings-and-connection-strings-work/
This should be safe enough - if the azure admin credentials are compromised, you lose everything, not just the connection strings - an attacker can simply use remote website debugs to get any information by stepping over the program, even if you had an encryption method / decrypt connection strings.
-Simon.
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