Basel airport transit to Lucerne?

There is a bus every few minutes from the airport to Basel SBB station. At the station there are luggage lockers of several sizes, although I don't know about the time limit on them. There is also a left luggage office, open normal hours, which might be better for longer term storage.

You can buy a through ticket from the airport to Luzern, since you can take any bus or train during the day, take as much time at Basel SBB before cntinuing on to Luzern.

There is a regular shuttle bus that goes from the arrivals level to Basel SBB station, bus No. 50.

If you are leaving your car at the French side of the airport, you need to cross to the Swiss side. Go up to departures level, cross to the Swiss side, and back down to Swiss arrivals.
The bus stop is just outside the arrivals exit. You must buy your ticket from the ticket machine before getting on the bus, the driver DOES NOT sell tickets. Price CHF 4.70. The machine takes Swiss Franks and Euros. The machine will sell you through tickets to anywhere in Switzerland, so you don't need to buy a new ticket at the station.

Bus leaves every 7½ minutes, extra buses at busy times.

Luggage storage: Yes, left luggage lockers. Go down the escalator from the main hall inside the station. Price varies by size, CHF 6 to CHF 9.

Regarding the time limit on lockers, I saw this on-line.

Generally, the maximum storage duration is 24h
However, you can extend the storage duration via supplementary payments before the 24h runs out, up to a maximum of 72h total
If a 24h period runs out without further payment, the contents will be locked and a red light on the locker should indicate that you need to pay again to access your contents
After the total maximum of 72h runs out, the contents of the locker will be claimed by the baggage office and moved to the lost & found, and you'll have to pick them up there (for a fee, I assume)

You have options.