Yes Helium Balloons can and do fly in the rain.
Although helium balloons can fly in the rain the rain causes the helium balloons to fly a bit slower than they would when the weather is sunny and not cloudy or rainy.
When a helium balloon is flying in the rain the volume of air that is displaced or the lift is unchanged.
That then causes the weight of the balloon to now be now be higher due to the surface water along with the collision velocity imparted by falling raindrops.
At events they usually do not fly helium balloons in the rain because the rain could potentially damage the Helium Balloons.