I'm going to assume you are going to host your WebAPI under IIS.
Your IIS Application is running under an Application-Pool. the Application-Pool runs under a certain Identity. The default Identity for an App-Pool is: ApplicationPoolIdentity (depending on your version of IIS)
See
http://blogs.iis.net/webdevelopertips/archive/2009/10/02/tip-98-did-you-know-the-default-application-pool-identity-in-iis-7-5-windows-7-changed-from-networkservice-to-apppoolidentity.aspx
If you follow the accepted answer to this question:
Login failed for user 'IIS APPPOOL\ASP.NET v4.0'
You'll see you need to give Sql-Server rights to the account running the App-Pool.
The quick way to test is to give the Identity running the AppPool.......the "db_reader" role. You won't be able to update, but you should see GET http actions work.
Your connection string would look like:
Server=myServerAddress;Database=myDataBase;Trusted_Connection=True;
"Trusted_Connection" means the code will try to connect to the database using the identity which is running the code.
But to summarized, figure out which Identity is running your WebApi code.....and then give some kind of rights to this Identity to your database inside of Sql-Server.
.......
Your other choice is to create a Sql-Server Login (username/password combination) and wire up your ConnectionString that EnterpriseLibrary uses to use this connection credentials information.
The below link calls it "Standard Security"
http://www.connectionstrings.com/sqlconnection/
Server=myServerAddress;Database=myDataBase;User Id=myUsername;
Password=myPassword;