Sabrina urges Linus to go to Paris to find love and life. But she says, he can't take a briefcase and "never an umbrella. There's a law."
The umbrella then becomes a sign of Linus' fuddy-duddy ways. He wants to be safe (in case of rain) and boring. But in Paris you have to toss aside that net, and frolic between the raindrops.
In the last scene, when Linus appears on the boat to go to Paris with Sabrina, he's carrying an umbrella. But just before he goes up to her, he hangs it from the pants of an unsuspecting man passing by. Without the umbrella, he's carefree, silly, and mischievous. He's the new, sparkling, lover Linus; he's no longer a boring businessman.