How to replace this connector
The power supply has the output label. Center positive for the barrel connector, 15V DC.

Yes, you would/could use your multimeter to measure the two wires, and determine which one is positive using the voltmeter feature (In DC mode if it has both AC and DC). Also use your multimeter's continuity test or ohmmeter feature to confirm which wire on the replacement connector goes to the center pin and outside of the replacement connector.