Under US tax law including Court Decisions, even though an asset was acquired many years before becoming a US taxpayer (Citizen or tax resident) the tax basis for determining gain or loss is the original cost basis of the property (or if inherited the fair market value of the property when it was inherited). That means when the asset does not get a revised basis on the date of becoming a US taxpayer and US taxes will be paid on appreciation of the asset prior to the date the individual became a US taxpayer.
What to do? Before becoming a US taxpayer, consider selling your highly appreciated assets to avoid paying US taxes on the gain that occurred prior to that date.
The one exception to the rule stated above, is when you Surrender your Citizenship or Long Term Tax Residency, for the purposes of determining if you have to pay an exit tax on form 8854, your tax basis IS the fair market value of the property or asset on the date you first became a Long Term Resident or Citizen.