If this is the whole trigger, I see no reason to involve a cursor.

CREATE TRIGGER trUpdateAnalVerification ON
[ERA_IL_Study_Analytes_Verification]
FOR UPDATE AS

DECLARE @rc int
SET @rc = @@ROWCOUNT

/*See if any rows were opened*/
INSERT INTO gerrystest 
VALUES (
 CAST(GETDATE() AS varchar(25)) + ', after open, ' + CAST(@rc as char(5)))

Also, a SQL Server trigger fires once for each data operation, no matter how many rows are modified. Change 1 row, the trigger fires once, 5 rows, the trigger fires once, zero rows - fires once.

-Mike