My friend has a similar problem. Her solution: when the chairs aren't being used she pushes them in and ties them together by slipping a looped rope over the tops of chairs on opposite sides of the table. It looks like this:
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Her son is almost 2 and still can't pull the chairs out on his own.

This problem makes me think of how they weigh down pinewood derby cars. Maybe you could buy tungsten or lead weights to be affixed to the wooden frame area under the seat. Maybe some fishing sinkers could be sewn or affixed with a hook and eye type of closure. If I were you I would begin my search in a hobby shop or fishing shop.

Or possibly make chair covers with a weighted bottom seam (you could make hidden pockets for the weights) this would solve the tipping issue as well as temporarily cover the chairs for when your child is young and messy.

I think that if they are light enough for your son to tip over, they probably won't do much damage if they fall on him (I am assuming this is your concern?) Just keep them tucked in and he might not even notice them for some time. (We were afraid my son would climb on the dining chairs, but so far he takes no interest).

Definitely making a note of this. I also (impractically?) want nice, 18/19th century style furniture AND to have kids. I am considering changing the upholstery when the day comes to a more spill-friendly fabric... perhaps faux leather. Probably gonna do two loops myself, one for each leg: 0===0. I imagine you can choose fabric that closely resembles the carpet/wall color in order to be less visually obtrusive. If you're having your chairs specially made, you can also choose heavy woods, my mom frequently chose furniture made from Roble, I think that's oak in English, but I'm not sure. Though I'm sure she must have looked silly in furniture stores shaking the beds and chairs to see how they'd stand up to us. I have a hall desk, and a chair sitting next to it for when I need to write thank you notes, sort bills, etc. When company with kids come, I can probably tuck the chair in and loop the opposite ends around the desk legs closest to the wall. As for when they're sitting on them, I can only imagine I'll be left to the mercy of (eventual) learned manners.

You could always wait and see whether it becomes an issue.
Others will have more creative ideas for good-looking solutions, but in case you want to consider a fast and non-permanent solution (albeit an ugly one), you could try strapping ankle weights (like the kind you'd use as exercise equipment) around the legs of the chair.

I would worry more about leaving those rope loops around for him to play with/get tangled in than the chairs. They are light, if he pulls them, they will fall. They will not hurt him and he will learn. Or you could try just telling him not to touch them...
Babyproof the things that can seriously hurt your child, let them learn to deal with the stuff that might merely bruise them or scare them. No need to wrap your house in plastic and tie everything down. Promise.

We don't have nice chairs, but we do have very light-weight chairs that we use for all our meals. Both my boys (3.5 yrs and 1.5yrs) have knocked them over a couple of times, but with no ill effects other than crying because of the loud noise. They've also tipped once each while sitting in the chairs, pushing off of the table with their feet. We haven't had a problem since, and they play with them all the time.

Barbed wire, you may need to wrap the legs in towels first so the wire doesn't scar the timber.

Hairymonster - haha!

I think holiday05 has the right idea. Remove anything that could kill them, babyproof anything that could cause serious injury, but cuts and bruises are learning experiences. Like hairymonster's suggestion though, that'd be a very quick lesson.

Don't worry about it.

Having the same Louis XVI chairs around my dining table, I've never thought they would be a problem. And they aren't. Don't worry about possible problems, just wait to see if there are actually any !

Also use low voltage wire! or high voltage if they are really bold!

You are right to be concerned. Also, that fabric does not look baby-friendly. Lead weights: not around babies, whose first instinct is to put things in their mouths.
Can you sew up some slipcovers that have fabric tabs attached that you can button around table legs? That would protect the fabric and keep the chairs upright without the danger of Junior strangling himself with rope.
It's true that there's only a small chance your child will strangle himself or cause brain damage by pulling a chair down on his head. Maybe one in a million. Still, I don't want you to be that one mom crying in the Emergency Room.

I took a look at the bottom of your chair legs and the design of the foot is part of the problem. Each of the legs sits on a nub and above that is an inverted bell which is not the best design for stability. It's like comparing a stiletto high heel to a bold wide heel shoe, there's no comparison for surface contact. The probable solution to your tipping problem is to temporarily widen the feet of your chairs to gain more surface contact. You may have to experiment with different materials. In order to not damage your chairs, I would recommend wrapping felt around the feet, this may not be the most attractive solution, but get some tennis balls, cut an 'x' and stick them on the chair legs, You may also try those swimming noodles or cylindrical styrofoam, you'd have to cut the noodle the length (probably 3"-4") and put a hole in the center for the leg to fit through making sure the foam is in contact with the floor. To keep the foam from slipping up, you may have to wrap additional felt at the top around the leg above the styrofoam. As for protecting the seats, plastic or vinyl chair covers. The DIY cheaper version, plastic sheeting, garbage bag, or a vinyl table cloth (there are tons of designs) fitted over your chairs. There are plastic seat covers available for purchase online, but most are designed to cover only the seat. Then over the plastic protection, put a pretty fabric chair cover - you can buy or make these yourself.