When I started bike-commuting with my road bike I developed chronic pain in my wrists. It probably took a year to go away, and now I have been free of complaints for several years. My wife just started bike-commuting with her road bike and she started to complain about painful wrists as well. I assume this has to do with holding weak wrists at an angle for an extended period of time. I was thinking about pointing her handlebars downward a bit, but am worried that she will slide off the grips mounted to the top of the handle bar (I don't know what those are called...) when she rides with her hands above the bar and has to brake abruptly. Something else we were thinking about trying is wrapping her wrists with elastic bandages, but we weren't sure if we were going to exacerbate or at least prolong the problem, since she would probably use her wrist muscles less.
I switched bikes a year ago, and I noticed on my rides (50km - 70km) at the time that I had some numbness / cramp / pain in my hands and wrists, it was so bad on some rides that I need two hands to change gear. The bike had slightly different geometry than my previous road bike. I now ride frequently 100km and even up to 180km with little or no pain.
Here's what I did and what changed:
1) I took the bike to my LBS and asked them if there was something they could do. The adjusted the shifters outwards so that I wasn't stretching so far to change gear. A good bike fit may be the only solution you need.
2) Good gloves with good padding in the right place, to prevent putting too much pressure on the nerves that run through your palm.
3) rotating hand position, on a road bike you have 3 (or possibly more ) hand positions. Use them. They are not always practical in every situation but you can often switch from the drops to the hoods or vice versa on the flat, and the hood to the tops or vice versa on climbs. This change in position will help with cramping.
4) finger clenches and stretches. As soon as I start to feel my wrists or fingers getting tired I take one hand at a time off the handlebars (where it is safe to do so) and clench and stretch my fingers 5-10 times.
5) Hold the handle bars loosely. My main problem was that I was gripping the handlebars way too tightly. If you relax your grip to the loosest possible you will reduce cramp and reduce the effect of vibrations from small bumps and rough road surfaces. This is especially important for long fast downhills. I would grip on for dear life (doing about 60kmph - 75kmph) clearly you want a harder grip as even small bumps can be dangerous, but squeezing very hard on the handlebars will give you cramp in no time at all. Also when braking on the down hills its better to pump the brakes rather that keep them on the whole time.
All this takes practice and I am sure that the muscles in my wrists got used to the abuse they were getting too. In the end I am fine.
Seems like you need a bike fitting for your wife. Depending on the length of her ride (probably 2-5 miles for people not used to longer riding), she may need to work up to riding the commuting length instead of just jumping to the full commute.
I'd suggest checking the saddle angle (it may be too downward), raising the handlebars if possible (this may or may not be possible), and checking her posture. Another thing is some bikes have very aggressive geometries for racing - you may just need a less aggressive geometry (some cyclocross bikes, touring bikes, etc.) or switching to something like a flat bar road bike (hybrid) or an old mountain bike (I prefer flat/riser bars for commuting under 10 miles, and a more upright geometry).
Sheldon Brown has a bit more things which may be useful, so I defer to him: http://sheldonbrown.com/pain.html
With drop bars you don't have a lot of options. Make sure she's comfortable riding on the hoods because that's IME the best position for wrist support. Often women have relatively shorter upper bodies than men of the same height, so on a bike that's the right height they're stretched forward. So she may be spending more time on the drops and flats than you expect, which will not help with wrist pain.
If she has or can fit flat bars, try bar grips with wrist support. The expensive-but-worth-it option is Ergon, who make a range of grips that are more or less all the same shape:
This stype of grip locks on to the handlebar (to stop it rotating) and supports the palm of your hand significantly more than a round grip does. There are cheaper versions of these around and most bikes shops should have a set. I found they did wonders for my comfort, and the silly-looking mini bar ends are great in traffic - they're just big enough that if my handlebar hits something they keep my fingers from getting crushed.
One slightly crazy option if you are locked into drop bars would be a flat bar with "drop bar ends". That would let you mount both ergon-style wrist supports and your existing drop bar brake/shifter setup.
Most common problem for wrist or shoulder pain is the saddle angle. The shoulders, arms, hands, and wrists need to hold your weight if the saddle points downwards. Hence, the saddle needs to be horizontally aligned. Here is a good manual on how to adjust the saddle: https://www.canyon.com/_en/supportcenter/article.html?supportcenter_categories_id=10&supportcenter_articles_id=148&type=rennrad
I would add to the above answers that wrist-wrapping is only a palliative; it may relieve the pain temporarily, but almost certainly won't solve the underlying problem. The pain will just come back, and the damage will worsen and become harder to heal.
I agree with Batman's suggestion of a bike fitting. Alternative handlebar designs (bullhorns, trekking bars) might also help.