The quickest way is to put it as an appsetting in your servers web.config and read the values at runtime directly from codebehind.
A web.config file is not viewable on a public facing webserver hosting IIS.
Web.config:
<configuration>
<appSettings>
<add key="ConnectionString" value="User Id=myusername;Password=myPasswordc;DataSource=my.datasource/>
</appSettings>
</configuration>
Code:
string CS = ConfigurationManager.AppSettings["ConnectionString"];
AzureService myAzureService = new AzureService(CS); <-- or similar
On a tangent:
Any sent TCP/IP packet sent over http (Client to server) could be sniffed if a hacker were to do so;
If you were sending authentication details from/to a client/server; any authentication detail might be sent to the Azure service in clear text.(over http)
If this was your intended setup, you should consider running your Cloud service over Https, creating a secure socket (tunnel) where any clear text authentication is protected by the SSL Tunnel