Database Connectivity using JavaScript.

You should use AJAX.

Create database connection method in Web service/ASMX and call that method in javascript. That web service reference should be added on script manager 's service reference tag.like

and after that call web service method in javascript.

you can also Jquery ajax means use jquery library and create JSON call to call connection method from web service or aspx.cs page which is declare as [web Method]
for example

$.ajax({

type: "POST",

url: 'Test.aspx/GetDataBaseConnection',//Here file path and method name

data: parameters,

contentType: "application/json; charset=utf-8",

dataType: "json",

async: false,

success: function (id)

{
//HERE METHOD will be call
}}