In Bitbucket Server, there is currently no way to disable or hide the ability to download archives from a repository in the Web UI.

This is the feature described here:
https://confluence.atlassian.com/bitbucketserver/download-an-archive-from-bitbucket-server-913477030.html

Example Use case:

Customer is hosted at an external provider and want to be able to restrict the ability of code to be downloaded to be only within their network while at the same time allowing for UI access over the web for devs to do code reviews and such.

The ability to restrict SSH and HTTP to the /scm path for the git clones to not be available except via the internal network interface can be accomplished, but the download feature via the UI provides for a full download of source which would in-turn prevent the customer from making the UI available over a web level connection.