Did geography and climate affect the development of slavery in the European colonies in North America? If so, how?

Yes, climate and geography affected the development of slavery in the European colonies of North America, although not by design. The original impetus for European exploration and colonization had been, according to King Henry 'the Navigator' of Portugal, to find new trade routes and supplies of gold required to pay his troops to prevent further encroachments by the slave trading Ottoman Empire on his people and land. The success of the Portuguese had encouraged the Spanish to follow suit.
The success of these Iberian efforts posed a threat to Protestant England. Philip V of Spain had already launched a massive naval Armada against England with the intention of deposing Queen Elizabeth I and restoring Catholic rule. Her successor, King James I, supported his own colonization efforts in America to help maintain English independence. The problem was the Virginia colony was a money pit and a death trap until John Rolfe (the English husband of Powhattan Princess Pocahontas) pioneered a new mild hybrid tobacco that flourished in the semi-tropical climate and appealed to European consumers.
This cash crop saved the English colony and launched an economic boom. Unfortunately planting, harvesting, drying, and packing was very labor intensive and the white indentured servants sent from England had a short life span in the semi-tropical south with its malaria-bearing mosquitoes. A recently captured Portuguese ship had disappointed its English captors by having no gold or silver on board. It was taken to Jamestown and its unfortunate cargo of Africans was sold as indentured servants to the Jamestown colonists who were in dire need of field hands. Thus began African indentured servitude in North America (slavery wasn't implemented until much later in the 17th c.).
Because the West African unfortunates hailed from a semi-tropical climate, they were disease resistant to malaria and other maladies that laid Europeans low in the south. This increased their value as laborers above European laborers and began the demand for more African laborers to work the tobacco fields. By the time slavery was finally legalized in the Virginia colony, Africans were deemed indispensable.
In the New England colonies, the soil was rocky and agriculture was not nearly so advantageous. Raising cattle, forestry, fishing, ship building and other industries occupied the settlers and there was little demand for field labor. Accordingly, slavery became further and further associated with the agricultural south where tobacco, rice, indigo, and cotton cultivation all required a large manual labor force in the field that could resist tropical diseases.