For the population data, see Goldewijk et al. (2010) for the methodology used in version 3.1 (the latest version as of April 2011). The primary sources of historical population counts at the national level for HYDE 3.1 include:
World Atlas of Population History by McEvedy and Jones (Hammondsworth: Penguin Books Ltd., 1978) (see this post for the application of this data in economics research)For the data source of sub-national level population counts, see the description in Goldewijk (2005).
A Concise History of World Population by Livi-Bacci (4th edition, Oxford: Blackwell Publishing, 2007)
Used by Bluedorn et al. (2009) and Fenske (2013), the latter of which summarises how the data is constructed in detail on pp. 1369-1370.
The cropland data is constructed as follows: For 2000, they rely on satellite image driven data compiled by Klein Goldewijk et al 2006. To extrapolate backwards, they assume (1) the surface area of cropland and pasture per capita increased or decreased over time at the country level; (2) there is no cropland if the area is urban, population density less than 0.1 person per km2, or annual mean temperature below zero degree Celsius; (3) the areas with the highest soil suitability fro crops (according to FAO-GAEZ data) or coastal areas and river plains first became cropland, (4) the areas with high slopes were the last to become cropland.