Depends on how the helium suddenly gains weight, and how far-spread that change is.
To make helium become denser, you must increase the weight of the individual helium molecules. Or you need to employ a special kind of magic that restricts the speed of Helium molecules to a fraction of their normal Brownian motion speed (i.e. slows them down and makes it so that they take up less volume for their weight).
If it is throughout the solar system, then the sun will certainly get problems -- it lives off fusing Hydrogen to Helium, after all. If helium is suddenly heavier, then that mass needs to come from somewhere, and that most likely will disturb the regular fusion process. Additionally, all helium that has already been produced and is massing in the sun's core, will become heavier and so the Sun's gravity will increase suddenly, and disturb its nice equilibrium. If helium is just slowed by magic, its volume will decrease suddenly, and so the sun's core will suddenly shrink. In both cases: There will at the very least be horrible solar flares for weeks, if the sun manages to stabilize. If not, we could be dealing with the sun suddenly going nova.
Earth only: If Helium is heavier than air, it will sink to the ground. There are 5.2 parts per million in the atmosphere; a rough estimate has 10km of atmosphere, so helium should pile about 5.2cm high on the ground. Since it will flow from higher ground down to sea level, rowing a boat on the ocean might become hazardous, and I am not sure what the fish in the water (and the ocean-dwelling mammals) will do.
Helium is used in the industry for supercooling magnets etc. It's got a melting point of -272°C, and slowing down its molecules will certainly raise that melting point by tens if not hundreds of degrees. As a result, any and all technical applications that use Helium for cooling, will suddenly run hot.
Helium is also a sideproduct of radioactive decay (alpha rays are helium nuclei). No clue how radioactive decay will be disturbed if suddenly the breaking-off helium cores are fifteen times heavier than normal (that mass needs to come from somewhere, after all). The alpha rays themselves should lose much of their already small penetrating power (can be stopped by a sheet of paper), but since they are a lot heavier they will probably cause a lot more damage when something like Iodine-13 is ingested. Additionally, whereever alpha decay takes place in an enclosed room, you are in serious danger of suffocating (it can't get out, so it will pile up like carbon monoxide).
Tumbling balloons like in your doodle will be the most harmless side effect of heavier helium by far. And, since many DIY youtube instructions for floating balloons all produce Hydrogen from a redox reaction (DO NOT DO AT HOME -- hydrogen is highly flammable, see the Hindenburgh!), there will be many balloons still afloat.
Unless you suddenly make Hydrogen denser, too.