Just wanted to point out that this has affected software revolving around finance for some time now. Historical data goes back into the 1800s and there are bonds with very, very long horizons (100-year or approaching it). The last time I checked on Bloomberg I believe the latest maturity date I could find in the system was sometime in 2089. I think the industries that have had to deal with this already will not be very affected by 2038.
edit: Link to C++ N3344 paper we wrote, describing how we represent dates (times/datetimes are similar): http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n334...
edit2: Anyone interested in examples -- on the early side:
ISIN GB0000436070, UK Gilt, issued 1853, perpetual annuity
On the future side, things have changed since I last looked. Example bonds with the latest dates I can find are: ISIN XS0560190901, Dong Energy A/S, callable 06/01/3010
ISIN US786581AA66, HSBC Holdings Luxembourg SA, callable 10/15/2997