默认我们的DataGridView cell是没有边框的，无伦是正常状态还是编辑状态，因为系统把TextBox的BorderStyle值改成了None，所以这里首先要把BorderStyle值改回来，那至于是FixedSingle还是Fixed3D,就得看你自己需要了,我这里改成了FixedSingle,方便绘制

主要是继承　DataGridViewTextBoxCell　类，需要在这里改点东西代码如下：

  public class DataGridViewTextBoxEditCell : DataGridViewTextBoxCell
    {
        public override void InitializeEditingControl(int rowIndex, object initialFormattedValue, System.Windows.Forms.DataGridViewCellStyle dataGridViewCellStyle)
        {            
            base.InitializeEditingControl(rowIndex, initialFormattedValue, dataGridViewCellStyle);
            TextBox textBox = this.DataGridView.EditingControl as TextBox;
            if (textBox != null)
            {
                textBox.BorderStyle = BorderStyle.FixedSingle;//改回边框
            }
        }

        protected override void Paint(Graphics graphics, Rectangle clipBounds, Rectangle cellBounds, int rowIndex, DataGridViewElementStates cellState, object value, object formattedValue, string errorText, DataGridViewCellStyle cellStyle, DataGridViewAdvancedBorderStyle advancedBorderStyle, DataGridViewPaintParts paintParts)
        {
            base.Paint(graphics, clipBounds, cellBounds, rowIndex, cellState, value, formattedValue, errorText, cellStyle, advancedBorderStyle, paintParts);
             
            //绘制边框
            Rectangle cellBounds1 = new Rectangle(cellBounds.X + 2, cellBounds.Y + 1, cellBounds.Width - 4, cellBounds.Height - 3);
            graphics.DrawRectangle(new Pen(SystemColors.WindowFrame), cellBounds1);
        } 
    }
    
     public class DataGridViewTextBoxEditColumn : DataGridViewColumn
    {
        public DataGridViewTextBoxEditColumn()
            : base(new DataGridViewTextBoxEditCell())
        {
        }

        public override DataGridViewCell CellTemplate
        {
            get
            {
                return base.CellTemplate;
            }
            set
            {
                // Ensure that the cell used for the template is a CalendarCell.
                if (value != null &&
                    !value.GetType().IsAssignableFrom(typeof(DataGridViewTextBoxEditCell)))
                {
                    throw new InvalidCastException("Must be a CalendarCell");
                }
                base.CellTemplate = value;
            }
        }
    }
    
    应用:
    private void Form1_Load(object sender, EventArgs e)
        {
            DataTable dt = new DataTable();
            dt.Columns.Add("name");
            dt.Columns.Add("name1");

            DataGridViewTextBoxEditColumn dataGridViewColumn = new DataGridViewTextBoxEditColumn();
            dataGridViewColumn.HeaderText = "columns";
            dataGridViewColumn.DataPropertyName = "name";
            this.dataGridView1.Columns.Add(dataGridViewColumn);

            DataGridViewTextBoxEditColumn dataGridViewColumn1 = new DataGridViewTextBoxEditColumn();
            dataGridViewColumn1.HeaderText = "columns";
            dataGridViewColumn1.DataPropertyName = "name1";
            this.dataGridView1.Columns.Add(dataGridViewColumn1);

            this.dataGridView1.AutoGenerateColumns = false;
            this.dataGridView1.DataSource = dt;
        }
        
        应该还有更多好的办法，希望大家能批评改正

C#控件-显示边框的DataGridViewColumn