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I read somewhere that you get exposed to a non-negligible amount of radiation from the sun when you travel by air. Would this affect an unborn child's fragile DNA?!

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Unless you're flight crew, frequent flyer or there's a solar storm, radiation is not a significant problem. The radiation comes from space in general and not significantly from the sun, so a night flight has pretty much the same radiation level.

Physicians can assure pregnant women who are concerned about radiation risks during flight that, for casual travel under normal solar conditions, the risk of direct harm from cosmic radiation is negligible. ... The dose limits [based on risk tables generated by the FAA] are set well below levels at which real harm has been demonstrated.

source: In-Flight Radiation Exposure During Pregnancy (PDF), American College of Obstetricians and Gynecologists.

This page contains the best Q&A about in-flight radiation concerns I've found. (Summary: It's generally safe.)

You can use this online calculator to determine how much radiation your flight would bring. To give you some perspective, this chart (here's another) shows that one flight across the USA brings you only 40 microsievert, which is one 25th of the yearly limit recommended by EPA, or just a few times more than a day's worth of background radiation on the ground. So if you fly 25 times in a year, you're still within safe limits.

Notable safety exception:

Radiation levels can be a lot higher during solar flares — bursts of electromagnetic radiation caused by disturbances in the sun's atmosphere. Fortunately, solar flares are rare and last only a short while. The National Oceanic and Atmospheric Administration posts solar radiation storm alerts on its web site. Some experts recommend that pregnant women check the site before flying and consider postponing their trip a day or two if a storm is forecast.

source: babycenter.com

But even solar flares aren't that dangerous, according to the Q&A page linked above:

Some measurements have estimated exposures of 100 times the usual exposure while flying which would bring your exposure in the range of a low-level exposure from many diagnostic radiological procedures. Even this exposure would not increase your risk for birth defects or miscarriage, which is 3% for birth defects and 15% for miscarriage for all healthy women when they begin their pregnancy.

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This radiation chart, while not made by scientists can help you get the gist of how much radiation can come from various sources. Notice that an airplane flight is in the first section. It has a lot of squares, which makes it looks scary. However, you'll notice in the second section that "living in a brick, stone, or concrete building for a year gives one almost twice the radiation that a plane trip does. And millions (billions?) of people live in concrete apartment buildings and have no problems producing healthy children. Hopefully this will help put you at ease.

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That is a very interesting chart! +1 Thanks! Now I am curious why brick/concrete would leak radiation. –  pixelfreak Jul 25 '11 at 22:39
    
The disclaimer cracks me up: "If you're basing radiation safety procedures on an internet PNG image and things go wrong, you have no one to blame but yourself." –  pixelfreak Jul 25 '11 at 22:45
    
Bricks contain tiny amounts of radioactive elements found naturally in all earth. Surrounding yourself with them such as in a house, slightly increases your exposure. –  Nat Aug 15 '12 at 21:29
    
The disclaimer that this is not made by scientists is misleading. The graphic has been made using available science on radiation doses, however the graphic has not been through a formal scientific peer review process. –  Nat Aug 15 '12 at 21:29

Airlines will not allow you to fly after a certain amount, and this varies airline by airline. We've been turfed off a flight at 34 weeks, so I'd advise you to contact the airline before booking a flight anywhere; travel agents will not ask you if you're pregnant.

Most of the studies on pregnancy and air travel have been done on female flight attendants. One study did show that there was a slight increase in first trimester miscarriage, but this was for the flight attendants who worked a greater number of hours.

Other studies were worried about in-flight radiation. This was also shown to have a slight increase in potential problems. However, these problems were more related to the length of time in the air, the route flown, and other flight phenomena.

There is minuscule radiation on the ground and it goes down the nearer you are to the sea-level. When flying and the jet is cruising at 33,000 feet (10,000 metres), the level of radiation exposure is 35 times that at sea-level. It is 64 times when the jet reaches 39,000 feet (13,000 metre), cruising altitudes seen in transcontinental flights. This might look alarming but in fact, for an occasional flier, this level of exposure is still perfectly safe.

With many young women working in the industry and many actually flying during that time before some are even aware of their pregnancy, it is somewhat reassuring that there is no real scientific evidence that their babies are adversely affected. Nonetheless, in Europe though not in the United States, air crew are classified as radiation workers and their exposure levels are monitored.

Some other points I have found:

• Avoid excessive flying. Although there are no hard and fast numbers. The flight attendants with the higher miscarriage rates flew on average 74 hours per month.

• Pregnancy is a thrombogenic condition. What this means is that, when pregnant, a woman’s risk of developing thrombosis is increased. That is an established fact.

This is a scientific study on the issue. I haven't read this cover to cover, but if you want actual facts and figures, I am sure they'd be in here.

In flight radiation exposure during flight

But there is one paragraph in the summary worth note:

Physicians can assure pregnant women who are concerned about radiation risks during flight that, for casual travel under normal solar conditions, the risk of direct harm from cosmic radiation is negligible. However, during some solar energetic-particle events, the dose rates at airliner altitudes may be significantly greater than usual. During these rare events, a pregnant woman should be advised to check the web site of the Space Environment Center of National Oceanic and Atmospheric Administration and to consider a brief postponement of her trip until the peak-dose period has passed.

One final note (and this from myself) is that if you're carrying twins, I would be worried about travelling at any stage over 30 weeks as they are born earlier. Just something to keep in mind. I fly quite extensively and my wife and kids have always joined me wherever I am, whenever they can. She's flown many, many times pregnant, and as long as you take the precautions you need to (more exercise, more fluids etc), you'll be grand.

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+1 for mentioning thrombosis. Depending on how far along she is, this is something that the asker should be worrying about more than radiation. –  afrazier Jul 19 '11 at 20:33
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It should be noted that airlines do not allow pregnant women to fly because of the risk of birth starting mid flight. An emergency landing would be a significant cost to the airline. –  Nat Aug 15 '12 at 21:35

For reasonable amounts of air travel, radiation levels are too low to be a concern (per the Mayo Clinic):

Decreased air pressure during flight may slightly reduce the amount of oxygen in your blood, but this isn't likely to cause problems if you're otherwise healthy. Likewise, the radiation exposure associated with air travel at high altitudes isn't thought to be problematic for most business or leisure travelers. There's a caveat for frequent fliers, however. Pilots, flight attendants and others who fly often may be exposed to more radiation than is considered safe during pregnancy. If you must fly frequently during your pregnancy, discuss it with your health care provider. He or she may limit your total flight time during pregnancy.

Always check with your physician before flying while pregnant. Many doctors will tell expectant mothers not to fly after 36 weeks.

The risks of miscarriage or flight-related complications are lowest between weeks 14 and 28. During the first trimester morning sickness and other pregnancy symptoms can make flying particularly uncomfortable.

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