The short answer here is you can't password protect LocalDb. LocalDb is designed to be a low friction database setup for development and testing, but it does not provide any of the advanced features of SQL or even SQL Compact Edition. From MSDN:
LocalDB is created specifically for developers. It is very easy to install and requires no management
and then there is this:
Moreover, if the simplicity (and limitations) of LocalDB fit the needs of the target application environment, developers can continue using it in production, as LocalDB makes a pretty good embedded database too
"pretty good" implies that while it will work, there may be better solutions. Essentially, the lack of security is a feature of LocalDB. It is designed to run as the currently logged in user, giving them full access.
If you need Database Security as a feature of your deployed application, but do not want to deal with the complexities of a full SQL installation, you should consider SQL Compact Edition.
See this article How to: Deploy a SQL Server Compact 4.0 Database with an Application. Deployment will vary depending on the deployment methods used by the rest of your application.
Entity Framework is an abstraction layer which is not specifically tied to a specific database technology. It will work identically with LocalDb, SQL Compact, SQL Express, MySQL, postgresql, etc... usually with nothing more than than the correct database driver and the correct connection string supplied.