There was a time before the late 1980s in which just about any certified airline could be chartered to fly U.S. military men and women. In 1985, after a terrible crash in Gander, Newfoundland, killed some 200 American soldiers, Congress created an extremely effective Department of Defense inspection agency to take over a function the Federal Aviation Administration had been fumbling. The result today is a substantial inspection and monitoring system considerably tougher than what the FAA could provide.
Of course, it is possible for a U.S. airline to be very safe and well-run and yet not on that list (perhaps because it doesn't want the government charter business). But since the ordeal of gaining DOD certification is a high hurdle, finding your charter airline's name on that DOD list is reassuring. You can view the list at http://www.af.mil/factsheets/factsheet.asp?id=173 .
Here's the main point. When you get interested in an advertised travel package, ask the seller what airline (or airlines) it uses, and if one of them is something unusual like Arrow Air or BubbasAir, ask if it's a DOD-certified carrier, or go check it for yourself. (By the way, Arrow Air is an approved carrier, while BubbasAir -- as far as we know -- doesn't exist.) Now, a caution here. The DOD list includes only U.S.-based airlines, so don't get nervous about a foreign carrier just because it doesn't show up there.
Many foreign airlines are easily recognized (British Air, Lufthansa), and some major carriers have been longtime charter specialists (Holland's Martinair, for instance, is an excellent, well-established carrier). Some, however, are seldom seen at U.S. airports, and frankly, even I have to research some of the logos I see popping up from time to time on charters.
If you don't recognize the name of the carrier the packager is going to use, research it on the Web starting with Google or Yahoo! (And just as with hotels, remember that even the most dysfunctional excuse of an airline can still spring for a slick Web site).
Also keep in mind that if you're heading to some far-flung corner of the planet, a strange-sounding airline with headquarters in that section of the world may be the only airline the packagers can charter. If you don't know the airline and you can't find any substantive information about its approach to safety, then the bottom line is simple: Regardless of price, if you're not comfortable with what you hear about the airline involved, it's probably best to move on to another package.
By the way, to find out whether a particular carrier has been involved in major accidents over the years, go to http://www.airdisaster.com/ and search the database with the airline's name. Keep in mind that almost all of the world's good and experienced airlines have had accidents. That's not the point. What you're looking for is a pattern of mishaps, especially in the past 20 years. That's by no means a perfect indicator, but it's a start.
Let me make this point again, because it's very important. Most of the world's major airlines using modern Boeing and Airbus equipment and a discipline we call "crew resource management" are more than reliable -- especially the major carriers of Europe, Australia, New Zealand and much of Asia. There are, however, some exceptions, and the best rule of thumb is to check them out as well as you can and then listen carefully to your own intuition. Differences do exist, and this is one place where "vive la difference" doesn't apply.