Before encrypting the connection strings, think about what you are trying to protect against by encrypting them. Your application will need access to the cleartext connection string in order and therefore will need access to the key. Therefore, an attacker who compromises your ASP.Net application will likely be able to steal the key and your protected connection string. So encryption is not really adding much benefit.
Instead of encryption, focus on how that file is handled by operations personnel and the file permissions that are applied in production. Only allow Read access to the ASP.Net worker pool account that your application runs as.