I'd actually recommend American dollars since not only are they widely accepted in Tijuana, but then you can even travel to the U.S. (assuming you have the paperwork/legal permissibility to do so).
You could do pesos, but I'd still recommend American dollars and download a currency exchange app on your phone to get a rough cost of things. Most places in Tijuana usually have a calculator on hand and will do the conversion at the register. It may sound like an inconvenience for you and/or them, but it never felt that way to me, money is money.
The other reason I prefer American dollars is because I already have a point of reference for costs in Tijauana based on dollars and not pesos: example: a taxi ride from the border to the hotel is 10 bucks including tip (maybe times have changed, but I doubt it's much more than $15). I usually give them $20 total because I'm a good tipper
What you'll find interesting is that there is no "taxi meter" (at least when I was there), and you often times want to ask the cost of the ride before committing to the trip.
Like say "Grand Hotel en Blvd Agua Caliente por favor" - and when he indicates he understands, you'll be like "si - quanto" (yes, how much)? Stick to the yellow cabs if possible (not that others aren't safe, but the yellow ones have the best reputation from my understanding), I've never been high balled but I can't imagine you being charged more than $20 for an airport ride to the hotel.
Yes you can use your debit/ATM card to pull out pesos, but again my recommendation is American currency. Up to you, no wrong way of doing it I suppose. I just don't think Euro currency is taken.
Note, I'm relying on information that is quite dated, but I have no reason to think any of the costs I mentioned have changed significantly; if anyone has any new insight, please chime in.