You can use ClientScript.IsStartupScriptRegistered
to check if it's already registered:
Public Class MyTextBox
Inherits TextBox
Private Sub MyTextBox_PreRender(sender As Object, e As System.EventArgs) Handles Me.PreRender
If Not Page.ClientScript.IsStartupScriptRegistered("AlertHelloWorld") Then
Dim alertHelloWorld = "window.alert('Hello World');"
Page.ClientScript.RegisterStartupScript(GetType(MyTextBox), "AlertHelloWorld", alertHelloWorld, True)
End If
End Sub
End Class