T-SQL: A proper way to CLOSE/DEALLOCATE cursor in the update trigger

Yes, use TRY/CATCH but make sure you deallocate etc after. Unfortunately, there is no finally in SQL Server.

However, I suggest wrapping this in another try/catch

CREATE TRIGGER trigger1 ON [dbo].[table1] AFTER UPDATE
AS 
BEGIN                           
    --declare some vars
    DECLARE @Col1 SMALLINT, @Col1 TINYINT 

    BEGIN TRY
        --declare cursor            
        DECLARE Cursor1 CURSOR FOR 
        SELECT Col1, Col2 FROM INSERTED                     

        --do the job
        OPEN Cursor1
        FETCH NEXT FROM Cursor1 INTO @Col1, @Col2

        WHILE @@FETCH_STATUS = 0
        BEGIN
            IF ...something...
                    EXEC myProc1 @param1 = @Col1, @Param2 = @Col2
            ELSE
            IF ...something else...
                    EXEC myProc2 @param1 = @Col1, @Param2 = @Col2

            FETCH NEXT FROM Cursor1 INTO @Col1, @Col2                               
        END
    END TRY
    BEGIN CATCH
        --do what you have to
    END CATCH

    BEGIN TRY
        --clean it up               
        CLOSE Cursor1
        DEALLOCATE Cursor1                                  
    END TRY
    BEGIN CATCH
        --do nothing
    END CATCH
END

Whether a cursor in a trigger is a good idea is a different matter...

You could use the CURSOR_STATUS() function.

if CURSOR_STATUS('global','cursor_name') >= 0 
begin
 close cursor_name
  deallocate cursor_name 
end

reference: http://msdn.microsoft.com/en-us/library/ms177609.aspx