Saving web search results for EANs to CSV
There are multiple things to tackle and improve, let's summarize.

Code Style

In general, try to follow the PEP8 style guide, you have multiple violations, like:


use of spaces around operators and blank lines
organizing imports


There are tools like flake8 or pylint which can analyze your code statically and report existing violations, make sure to give them a try.

Variable naming is also a huge issue in the presented code - variables like i or my_dictionary are meaningless - make sure to give your variables meaningful names to improve readability. Remember: code is much more often read than written. 

Code Organization

Currently, you are putting everything into a single program without any organization - think about logically splitting it into functions. You should also put your main execution logic into the if __name__ == '__main__': to avoid the code being executed on import.

Making the code Pythonic


you can use a dictionary comprehension with enumerate() to define your new_dictionary dictionary:

with open('EAN.txt') as f:
    new_dictionary = {
        'sequence_{}'.format(line_number): line
        for line_number, line in enumerate(f)
    }


On the second though, you don't actually even need a dictionary here - the keys are useless in the main execution logic of the program. A list of search queries should be sufficient enough.
you don't need read().splitlines() and can directly iterate over the file object lines - this will avoid having an extra list of lines created in memory and allow to get every next line in a "lazy" fashion
use print() as a function for Python-3.x compatibility


Performance and Web-Scraping related Improvements


First of all, you don't need to save the page source into a file and then reading that file to parse with BeautifulSoup - you can avoid that step altogether by feeding mechanize Browser's page source into BeautifulSoup directly.
You should also be able to initialize the Browser once and re-use for all the subsequent requests.
Also, you may improve on HTML parsing speed by using lxml as an underlying parser for BeautifulSoup - this requires lxml to be installed though.
I would also use a .product-title CSS selector to match product titles.
I would just collect the results and then write at the end instead of dealing with re-opening the resulting CSV file in the loop.
Performance-wise, this is also a good use case for a SoupStrainer which will make BeautifulSoup parse only the desired parts of a document - in your case the product links




Here is the code with some of the improvements applied:

import csv

from bs4 import BeautifulSoup, SoupStrainer
import mechanize


URL = "https://mysite"


if __name__ == '__main__':
    print("Running...")

    results = []
    with open('EAN.txt') as f, mechanize.Browser() as browser:
        browser.set_handle_robots(False)

        for search_query in f:
            browser.open(URL)

            browser.select_form(id="searchForm")
            browser["q"] = search_query
            response = browser.submit()

            soup = BeautifulSoup(response, 'lxml', parse_only=SoupStrainer(class_='product_title'))

            product_title = soup.select_one('.product-title')
            product_title_text = product_title.get_text(strip=True) if product_title else None
            results.append([search_query, product_title_text])

    # dump results
    with open('Results.csv', 'ab') as csvfile:
        writer = csv.writer(csvfile, delimiter='|', quotechar='|', quoting=csv.QUOTE_MINIMAL)

        for search_query, product_title in results:
            result_label = "Approved" if product_title else "Rejected"
            writer.writerow([search_query, result_label, product_title])


Might not work as is, since I don't have a way to test it. There are also some assumptions there - like the presence of a only one "product title" on a search result page per form query. But, at least, I hope that will give you something to start with.

And, I am not sure you need that np.genfromtxt() since currently you don't even read the results of this function call - hence removed that part in the posted code above.