SQL Server SSL Connection - Performance
The connection encryption happens at the SQL Server instance level, not at the level of individual databases. 

You can have encryption turned on at the instance, in which case every connection to it will be encrypted, or you can set a client to require encryption. Keep in mind every instance such a client connects to will have to have a certificate which the client trusts. Some light reading: https://support.microsoft.com/en-us/help/316898/how-to-enable-ssl-encryption-for-an-instance-of-sql-server-by-using-microsoft-management-console 

So to answer your question directly, yes, if one client connects to an instance with encryption and others don't, that one client will require more processing power than the others. Having said that, unless there's a configuration problem or the server is having processing capacity issues, nobody should notice.