To compare on a "per captia" basis, means to use "per captia" figures for your comparison. You can similarly say

I will compare the drinks on a "sugar content" basis

This means I will use the sugar content figures to compare the drinks.

The claim is that for "Hotspots" like New York, the (number of tests in New York) ÷ (number of people in New York) is greater than (the number of tests in South Korea) ÷ (The number of people in South Korea). I've no idea if that claim is true.

This compares New York with Korea on a "per capita" basis.

To compare on a "per capita" basis, means to use "per capita" figures for your comparison. You can similarly say

I will compare the drinks on a "sugar content" basis

This means I will use the sugar content figures to compare the drinks.

The claim is that for "Hotspots" like New York, the (number of tests in New York) ÷ (number of people in New York) is greater than (the number of tests in South Korea) ÷ (The number of people in South Korea). I've no idea if that claim is true.

This compares New York with Korea on a "per capita" basis.