When I'm on a trip involving airplanes, if I have a long connection between flights, or arrive especially early to my first airport, I will check and see if there is a chapel around. Even if you are not religious, the quiet can be pretty nice.
Here are some notes I have compiled about the various airport chapels I've been to. This page will be updated periodically, as I have occasion to connect through more airports.
Dallas-Fort Worth (DFW)
Operated by the Airport Interfaith Chaplaincy; open 24/7.
The only airport I've been to where there are apparently multiple chapels in different terminals. I visited the chapel in Terminal C, which had plenty of space, some nice stained glass (no patterns, just colored squares), and was tucked away in a relatively quiet corner of the terminal.
Some other folks (I think airport employees) visited the chapel while I was there, which was great to see!
Various religious books are available; most interestingly, there was a "Proposed Book of Common Prayer" released before the 1979 BCP was finalized. I had never seen this edition before, but I don't think there were any major changes between it and the approved 1979 one (I didn't notice any praying Rite 2 Evening Prayer, at least!).
New Orleans (MSY)
They call it a "meditation room" here. It's post-security, but was hard to find and felt kind of cramped. I wouldn't recommend it.
Boston (BOS)
"Our Lady of the Airways" is actually the first airport chapel built in the US! It's before security in Terminal C. It feels very Catholic because it is; there are lots of pews, Mass is held on Sundays, and the only prayer book available is a Misalette for praying along at Mass (I didn't even see a Bible anywhere). It does have operating hours though they seemed kind of loose.
Since it's before security, it is hard to use if you have a connection or if security lines look long. But it's quiet, big, and definitely worth checking out!
Washington National Airport (DCA)
The chapel at DCA is post-security, near the A gates. If you're walking towards the A gates from the opposite side of the airport, turn right into the hallway it looks like you're not supposed to be in, instead of left towards the gates.
It does have operating hours (6am-9pm) but the noise isolation is phenomenal (I think they are running a fan or white noise machine inside the space). Some prayer books were available.
There are no windows or other interesting features, but there is ample space available. Definitely worth visiting if you have time.
Charlotte (CLT)
The chapel is on the second floor in the atrium area, post-security. The wayfinding is quite good so it's easy to find.
The noise isolation is not great; you can hear most everything going on outside, and this is not a quiet area. It's a small space, but not as cramped as MSY. Limited prayer books were available. I found it just a little lacking, but perfectly serviceable.