A user is not allowed to resize a TextBox control vertically. The height of a TextBox is locked to the ideal height that a textbox should be.
What's more, Visual Studio doesn't even provide you vertical drag handles:

How do i provide the same mechanism on my UserControl?
I will elaborate on Hans' comment. You can associate specialized code (called a Designer) with a UserControl so that when it is placed on a form in Visual Studio, the user is limited in how they can configure your control.
Add a reference to System.Design in your project.
Use the following sample code:
[Designer(typeof(FixedHeightUserControlDesigner))]
public partial class FixedHeightUserControl : UserControl
{
    private const int FIXED_HEIGHT = 25;
    protected override void OnSizeChanged(EventArgs e)
    {
        if (this.Size.Height != FIXED_HEIGHT)
            this.Size = new Size(this.Size.Width, FIXED_HEIGHT);
        base.OnSizeChanged(e);
    }
    public FixedHeightUserControl()
    {
        InitializeComponent();
        this.Height = FIXED_HEIGHT;
    }
}
public class FixedHeightUserControlDesigner : ParentControlDesigner
{
    private static string[] _propsToRemove = new string[] { "Height", "Size" };
    public override SelectionRules SelectionRules
    {
        get { return SelectionRules.LeftSizeable | SelectionRules.RightSizeable | SelectionRules.Moveable; }
    }
    protected override void PreFilterProperties(System.Collections.IDictionary properties)
    {
        base.PreFilterProperties(properties);
        foreach (string p in _propsToRemove)
            if (properties.Contains(p))
                properties.Remove(p);
    }
}
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