public partial class Alerts : Form
   {
      #region Private Fields

      private static BackgroundWorker bgw = new BackgroundWorker();
      private int _period = 5000;
      private System.Threading.Timer _timerThread;

      #endregion Private Fields

      #region Public Constructors

      public Alerts()
      {
         InitializeComponent();
         bgw.DoWork += bgw_DoWork;
         bgw.RunWorkerCompleted += bgw_RunWorkerCompleted;

         _timerThread = new System.Threading.Timer((o) =>
         {
            // Stop the timer;
            _timerThread.Change(-1, -1);
            // Calls UpdateAlerts() that updates a datagridview with the mysql data

            bgw.RunWorkerAsync();

            // start timer again (BeginTime, Interval)
            _timerThread.Change(_period, _period);
         }, null, 0, _period);
      }

      #endregion Public Constructors

      #region Private Methods

      private void bgw_RunWorkerCompleted(object sender, RunWorkerCompletedEventArgs e)
      {
              DataGridView1.Invoke(new System.Action(() => 
    {DataGridView1.DataSource = e.Result;}));
      }

      private void bgw_DoWork(object sender, DoWorkEventArgs e)
      {
         string constring = "datasource=localhost;port=3306;username=root;password=******";
         MySqlConnection conDataBase = new MySqlConnection(constring);
         MySqlCommand cmdDataBase = new MySqlCommand(" select * from shopmanager.alerts ;", conDataBase);

         try
         {
            conDataBase.Open();
            MySqlDataAdapter sda = new MySqlDataAdapter();
            sda.SelectCommand = cmdDataBase;
            DataTable dbdataset = new DataTable();
            sda.Fill(dbdataset);
            BindingSource bsource = new BindingSource();

            bsource.DataSource = dbdataset;
            sda.Update(dbdataset);
            e.Result = dbdataset;
         }
         catch (Exception ex)
         {
            MessageBox.Show(ex.Message);
         }
         conDataBase.Close();
      }

      #endregion Private Methods
   }

您不必每次启动计时器时都订阅dowork事件。
您永远不会订阅runworkercompleted事件。
您应该将grids数据源分配给e.result，而不是不能由bgw更新的私有属性
不确定这将工作无论如何，你可能想看看调用。实际上，代码将抛出一个跨线程操作无效或目标调用异常。您需要在网格控件上使用invoke。

我将对您的代码进行一些更改：
使用 Timer 从windows窗体工具箱而不是 System.Threading.Timer （它的设计是为了在windows窗体中更好地工作）。将其拖到窗体上，然后将事件处理程序附加到其 Tick 方法来获取定期更新。
连接 BackgroundWorker 事件处理程序只有一次。
将backgroundworker组件从设计器工具箱拖到窗体上，以便为您管理其生存期；不要使用 static 现场。
使用 using 对账单 MySqlConnection 清理它使用的资源。
使用示例方法（不是 static )因此，您可以引用 Alerts 班级。
举个例子：

public partial class Alerts : Form
{
    public Alerts()
    {
        InitializeComponent();

        // set up BackgroundWorker once
        backgroundWorker1.DoWork += OnDoWork;
        backgroundWorker1.RunWorkerCompleted += OnRunWorkerCompleted;

        // set the timer and start it running
        timer1.Interval = 5000;
        timer1.Start();
        timer1.Tick += (o, e) => backgroundWorker1.RunWorkerAsync();
    }

    // This method will run in the background every five seconds. It can't
    // access parts of the form since it is on a background thread.
    private void OnDoWork(object sender, DoWorkEventArgs e)
    {
        using (MySqlConnection conDataBase = new MySqlConnection(connectionString))
        using (MySqlCommand cmdDataBase = new MySqlCommand(" select * from shopmanager.alerts;", conDataBase))
        {
            conDataBase.Open();
            MySqlDataAdapter sda = new MySqlDataAdapter { SelectCommand = cmdDataBase };
            DataTable dbdataset = new DataTable();
            sda.Fill(dbdataset);

            // return the results to the main form thread through this property
            e.Result = dbdataset;
        }
    }

    void OnRunWorkerCompleted(object sender, RunWorkerCompletedEventArgs e)
    {
        // This will run on the main form thread when the background work is
        // done; it connects the results to the data grid.
        dataGridView1.DataSource = e.Result;
    }
}