Your girlfriend should have no fears at all. The technology on the bridge is compared with NASA. I wish we had this data available when I flew as a crew member in the war zones (60's), I know dating myself. Cruising today is very safe. The electronics, crew spotters and all the information sent to the bridge is overwhelming. The first and foremost item is safety of the passengers, crew and ship.
As for how close the ship comes to an iceberg is the discretion of the Captain. The closet I've heard and been is 1/2 mile, and you would swear you were only a small bit away.
No Captain will put the ship and all the people in harms way.
Tell her the fears have no merit!!
Enjoy Alaska, it is a wonder no one should miss.