Enable and disable textbox using dropdown

Simply by checking for it's disability. If it is disabled then skip validation else force to check.
Example:

function validateControls()
{
  var txt=document.getElementById("<=txtboxName.ClientID%>");
  if(txt.disabled==false)
  {
    if(txt.value.trim()=="")
    {
        //alert msg
        return false;
    }
    else
    {
       return true;
    }
  }
}

Or,
Checking for the dropdownlist selected value.
Example:

function validateControls()
{
  var drp=document.getElementById("<%=drpName.ClientID%>");
    if(drp.value.trim()=="Other")
    {
        //alert msg
        return false;
    }
    else
    {
       return true;
    }
}

Hope, it helps :)