Hi, you can edit this in the base.xml of your web module (path: addons/web/static/src/xml/base.xml). The DB list is defined as

<t t-name="Login.dblist">
<select name="db">
    <t t-foreach="db_list" t-as="db">
        <t t-if="selected_db === db">
            <option t-att-value="db" selected="true">
                <t t-esc="db"/></option>
        </t>
        <t t-if="selected_db !== db">
            <option t-att-value="db"><t t-esc="db"/></option>
        </t>
    </t>
</select>
</t>

Now replace that selection "db" with a text box with the same name of the selection field as

 <t t-name="Login.dblist">
<input name="db" type="text" value="" autofocus="autofocus"/>
 </t>

EDIT :

Then in chrome.js (addons/web/static/src/js/chrome.js) comment the on_db_loaded method (line 676-690) , which is

on_db_loaded: function (result) {
    var self = this;
    this.db_list = result;
    if (!this.selected_db) {
        this.selected_db = result[0];
    }
    this.$("[name=db]").replaceWith(QWeb.render('Login.dblist', { db_list: this.db_list, selected_db: this.selected_db}));
    if(this.db_list.length === 0) {
        this.do_action("database_manager");
    } else if(this.db_list.length === 1) {
        this.$('div.oe_login_dbpane').hide();
    } else {
        this.$('div.oe_login_dbpane').show();
    }
},

comment the above full code.

Then save the files and reload your web page, you don't need to restart the server or upgrade the module as you are editing on files in static folder. Now the dropdown will be displayed as a text field as

EDIT 2

@Sureka

After you have changed the database text field to the center of the page comment the code in chrome.js (addons/web/static/src/js/chrome.js) as shown below:

database_selected: function(db) {
//        var params = $.deparam.querystring();
//        params.db = db;
//        this.remember_last_used_database(db);
//        this.$('.oe_login_dbpane').empty().text(_t('Loading...'));
//        this.$('[name=login], [name=password]').prop('readonly', true);
//        instance.web.redirect('/?' + $.param(params));
    },

which is just above the on_db_loaded method. Then refresh the login page and try to login.