Suggest how to extract method
Something along the lines of this. Generics aren't that difficult. Just replace every occurance of 'some desired type' with the generic identifier.

private void AddUpdateRow&lt;T&gt;(BindingSource bindingSource, List&lt;T&gt; cachedList)
    where T : ITransmitDocId
{
    try
    {
        bindingSource.EndEdit();
        T t = bindingSource.Current as T;
        if (t.TRANSMIT_DOC_ID == 0)
        {
            if (!cachedList.Contains(t))
            {
                cachedList.Add(t);
            }
            return;
        }
        context.UpdateObject(t);
    }
    catch (Exception ex)
    {
        logger.ErrorException(string.Empty, ex);
    }
}


The where constraint in the function definition should indicate a common interface/class in which TRANSMIT_DOC_ID is defined.