The 1-1/2” drain from the sink currently runs through the back of the cabinet and down into an interior wall to the basement. The new location will be an outside wall. There is currently no visible drain vent in place. What would be the proper way to vent the line in its new location?
There isn't only one proper way. If it's convenient to do so, extend the vent straight upward. If there is attic and roof immediately above then you could make a new roof penetration for the vent and detail with a pipe jack flashing in the usual way. If there's another story of the building above, you could get into that ceiling/floor space and take the vent anywhere - continue upward, route to meet with the existing vent, etc.
If taking the vent up is problematic, route it downward instead. It could be arranged as for an island sink as explained at Family Handyman (they mention two methods: loop as per the image below, or use an air admittance valve).