There seem to be a couple possibilities here, and not all of them constitute a "mistake" on your part.
One concerns the nature of the "purple" color. It could be possible that your indicator is correctly responding to the acidic conditions of your solution. I have attached an image, and dependant upon personal perception you may just be seeing a red that you are misidentifying.
If the color of your solution is indeed truly purple, a primary concern could be the cleanliness of the glassware you used. Methyl red produces a color ranging from red to yellow, and it not capable of producing purple. None of the other components of your solution, HCl and borax, nor any borate that could be produced in their acid-base reaction present either a purple color, nor a blue color which would shift the red of the indicator towards purple. Given this, it could be possible that your glassware was contaminated with another substance, and this is resulting in the purple color you are observing.