There are a couple of Anthem articles, but I can't remember where (CodeProject may have one). In theory, Anthem is very straightforward: to Ajaxify an ASP.NET web control (i.e. Label or TextBox), you just substitute the Anthem version. For example, if you want to turn an ASP.NET Button postback into an Ajax callback, just replace <asp:Button> with <anthem:Button>.
Not all Anthem controls make callbacks (or postbacks), but all Anthem controls are updatable. That means all Anthem controls can be updated on the client when the callback response is parsed. To perform this trick, all Anthem controls encase the underlying ASP.NET control elements inside of a <div> (for controls that normally render block elements) or a <span> with a special id. If the callback response includes an updated version of underlying control, then the entire innerHTML of the wrapper <div> or <span> is replaced. That is why you are loosing your client side updates.
I don't have an easy answer for you. Obviously, you could just avoid updating Anthem controls with client-side script, but I'm sure you will run into circumstances when you want to update the control from both the server and the client. In those cases I think you will need to reconcile the changes on the server.
For example, in my Shipping Rate Provider Suite I use a combination of Anthem and Yahoo! User Interface controls. In some situations I need to reconcile changes that were made via YUI with changes that I am making on the server. When that happens, I add code to the server-side that re-applies the client-side changes. I do this using Anthem.Manager.RegisterClientScript methods or Anthem.Manager.AddScriptForClientEval. These methods inject javascript into the callback response which are evaluated on the client when the callback returns to the client.
There are other .NET Ajax libraries that do not replace the entire control HTML during an update. They just update the properties that were changed during the callback. Those libraries would probably work better for you since they would not stomp on your other client-side changes (or at least not as often). I'm pretty sure MS Ajax rewrites the entire control, just like Anthem, so it is not such as good solution for you.