On our first trip to Europe together 40 years ago my husband and I arrived by train to Venice at about 5 pm on May 1 completely oblivious to the holiday. No boats running. Shops closed. We walked a mile over little bridges hauling our giant bags (before rollies and of course first trip so we overpacked). but it was 70F and sunny and beautiful and I still remember it well.
Transport in big cities like Paris will be running but I would assume everything else is closed and in smaller towns especially don't assume anything. Make transfer arrangements if needed ahead. and have a plan B and C for food etc. If you are staying in a hotel ask them about local dining arrangements. If an apartment, see if the landlord will stock the kitchen with basics for a fee.
We have enjoyed the labor demonstrations in Paris on May 1 for many years and there are food trucks there. But the tone has really changed. For years it was jolly labor songs including Woody Guthrie music and families pushing strollers as well as unions with their flags. This past year few kids and lots more black block goons. We got caught in the tear gas last year. It is a handful of thugs who spoil it and it is easy to steer clear as they are garbed in black and masks. We were stupid to not have gotten further away but the black block marchers were just marching when we were taking pictures and observing -- things changed fast. One year when the black block started their mischief, the cops deftly shifted the thousands of marchers down the river to the next bridge and circumvented the violent group while not disrupting the parade.