I never did find a way of making Internet Sharing stay on, but I did come up with a way of achieving the same result. With a quick bit of Ruby, I whipped up this solution.
Load IRB by opening terminal and typing irb
Copy and paste this into irb, and hit Enter to activate.
while(true);sleep 15;puts Time.now.to_s + ": " + `launchctl load -w /System/Library/LaunchDaemons/com.apple.InternetSharing.plist 2>&1`;end
As it stands, the code will re-activate Internet Sharing every 15 seconds. Now, that may seem like too short of a period, but at 15 seconds there's a good chance that the connection will be re-enabled before any connection drops occur. You can change 15 to any number of seconds that you like.
This will result in a log such as:
Mon Jul 25 09:37:07 -0400 2011: com.apple.InternetSharing: Already loaded
Mon Jul 25 09:37:22 -0400 2011: com.apple.InternetSharing: Already loaded
Mon Jul 25 09:37:37 -0400 2011: com.apple.InternetSharing: Already loaded
It's important to remember that closing the window will terminate the script (Internet Sharing won't re-activate every 15 seconds), so if you need to close the window, consider using screen which is included on Macs.
On the Mac:
-- Use 128-bit encryption. (Windows has 64 and 128, Apple has 40 and 128.)
-- Use a 13-character password.
On the PC:
-- Under "Wireless Security Type", select "Use Static WEP keys".
-- And finally, under "Access Point Authentication:", select "Shared".
Best Answer
It is not possible to use OS X's internet sharing feature to create a WPA- or WPA2-protected network [through v10.7 Lion, see below]; WEP (40- or 128-bit) or no encryption at all are the only available options.
Note that the original title on this question ("Airport supports WEP but not WPA?") was a bit misleading, as this is only a limitation on wireless networks created with internet sharing. Mac computers can join WPA and WPA2 wireless networks, and have been able to for years and years. Similarly, Apple's Airport base stations (except for early models) can create WPA and WPA2 networks. It's only when the Mac is acting as a base station that this limitation exists.
UPDATE: In OS X 10.8 (Mountain Lion), the normally-visible security options changed to WPA2 Personal or no encryption. WEP is still available in 10.8, but only if you hold Option while clicking on the Security pop-up menu.