Boil the potatoes in their skins until very tender, then drain. As soon as they are cool enough to handle, slip off the skins and mash or put through a potato ricer. Put in a large bowl and let cool completely.
Set up a large skillet and a large plate lined with a double layer of paper towels for draining the croquettes.
To the potatoes, add the salt, corn kernels, and half of the basil. Stir to mix well. Form 1/4 cup portions, shape into patties, and coat with panko. Place on a plate and chill for at least an hour, or until time to cook.
Place the large skillet over medium-high heat and let heat for a few seconds, then pour in oil to 1/4-inch deep. Let the oil heat to a "shimmer" and place one croquette in the pan. If it sizzles, add as many as you can fit without any of them touching. Cook, using a metal spatula to carefully turn them as they brown.
Drain the finished croquettes on the paper towel lined plate.
In a small bowl, stir the mayonnaise with the remaining basil.
Serve croquettes with basil mayo.