By using custom validator.

Client Side validation:
A Javascript function can be written to disable/enable the 
text box and it can be called under 
DrowpDownList_SelectIndexChanged event. We can provide an 
if condition and get the required change in javascript 
function.

or else

Server side validation:
we can make the dropdownlist 'AutoPostBack' property true.
and under the DrowpDownList_SelectIndexChanged event, we 
can disable/enable the textbox.

by using javascript function can be written to 
disable/enable the text box and it can be called under 
DrowpDownList_SelectIndexChanged event.

U can do it By Adding a Required validator To the textbox 
with enabled is false...

no in the Drop downList selected IndexChanged make it 
enable True... this will solve the Isssue

just write a javascript function like this
on button click or on onkeypress event of text box

function validate()
{
if(document.getelementbyid("ur dropdownlist").value=="other")
{
if(document.getelementbyid("ur textbox").value=="")
{
alert("please enter city name");
return false;
}

}

}

and call like this onkeypress='return validate()'

On_page load
{
RequiredFieldValidator1.Enabled = false;
}

if (DropDownList1.SelectedItem.Text == "Other")
        {
            RequiredFieldValidator1.Enabled = true;
        }
        else
        {
            RequiredFieldValidator1.Enabled = false;
        }

Hi Friends,

I hope the below solution helps....

1. Please add the required field validator for the texbox.
2. In Page_Load method, we can add the below statement

Dropdownlist.Attributed.Add
("OnChange","javascript:ValidateOtherCity();")

Add the below javascript function on the aspx page

function ValidateOtherCity()
{
  var other = window.document.getElementById
("DropdownlistName");

if(other.value == 'Other')
{
  ValidatorEnable(RequiredFieldValidatorName,true);
}
else
{
 ValidatorEnable(RequiredFieldValidatorName,false);
}
}