You will still be able to open the web map in desktop. The data, however, is referencing your service, so it's not actually getting downloaded, assuming you are using a map service.
There are two types of services, map and feature services. Map services are drawn on the server and generate an image that are drawn by the client application. Feature services return the actual geometry of features for clients to draw. One of the primary purposes of creating a feature service. One of the main purposes of feature services is to support editing.
Your post indicates you are using a feature service, or at least looking at the capabilities of the feature service. If you enabled feature access, then hiding the shape field will have no impact. Feature services, by nature, return the data to the client.
If you don't need to support editing (actually, I'm not sure what the basic edition supports), you probably just want a standard map service with feature access disabled on the Capabilities tab of ArcGIS Server Manager. The Mapping capability is where you could disable the Data operation. However, hiding the shape field should prevent the actual data from being accessed, while still supporting attribute queries.
hope this helps,
Mike