My brother-in-law bought some Japanese stocks in Japan during the
Korean War.  He just cashed in the stock and $170,000 was deposited in
his US bank.  He wants to know if he has to pay a capital gains tax. 
If so there is a second question: how does he calculate his gain since
he has few if any records of the cost basis?

Yes, the foreign stock your brother-in-law sold is a capital gain and
is taxable.  Any bank transaction of over a certain amount ($10k if I
recall correctly) is reported to the government, so if the income
isn't declared, it is entirely within the realm of possibility that
someone will come around asking questions about the income.

If you do not have documentation to prove the cost basis you must set
the basis as zero and count the whole of the sales proceeds as a
capital gain.  There may be some other forms that must be filled out
since this is a foreign holding.

More more information:
* Read IRS publication 550 (Investment Income and Expenses):
http://www.irs.gov/pub/irs-pdf/p550.pdf
* Also if foreign tax was paid on the sale see IRS publication 514
(Foreign Tax Credit for Individuals):
http://www.irs.gov/pub/irs-pdf/p514.pdf to see if your brother-in-law
qualifies for that credit on the foreign tax paid.