It looks to me like you're checking all connections to the database. If there are many such connections for different users, that could be a drag on the trigger.

So, I suggest using EXISTS() instead of COUNT and, if possible, using @@SPID instead of db name to use rows from sysprocesses. I'm sure you know that sysprocesses is deprecated and that sys.dm_exec_sessions is recommened now instead.

IF EXISTS(SELECT 1

FROM DatabaseLogonAudit.dbo.AuditDatabase ad

INNER JOIN master.sys.sysprocesses sp on sp.spid = @@SPID

INNER JOIN master.sys.dm_exec_connections ec ON ec.session_id = @@SPID

WHERE sp.loginame = @LoginName

AND ad.DatabaseName = @DBName

AND ad.DBLogonAuditRequired = 1

AND ec.Encrypt_option <> 'TRUE')