As others have mentioned, this is really only a problem in some very specific markets (San Francisco, Silicon Valley, Manhattan). It's not really an issue for most of the U.S.
That being said, we have to weigh the pro's and con's of either limiting or levying heavy tariffs on foreign purchases (not just Chinese, that'd be absurd) of property in the States.
The pro:
1. Capital flows into the U.S. Those foreign landowners not only pay the locals they purchase the property from a sizable chunk of cash, they also pay sales tax (where applicable), property tax, HOA, etc. All of that money goes towards the local economy in one form or another.
2. Wealthy foreigners now have a stake in U.S. soil. This gives them an incentive to favor policies both by their government and ours to protect their investment. Which means they'd be pretty unhappy about policies that'd crash housing prices in the U.S.
3. Less capital is invested in their native country. Economics amongst nations is, in many ways, a competition. The economy that's doing the best is the economy that's doing the least-bad. If foreign capital flows into the U.S. and not into the native nation, it means that there's less capital to prop up the native nation's markets and the U.S. is more attractive to park your $$. This makes the dollar stronger which makes foreign goods cheaper. It also allows the U.S. to have more money to invest into other ventures (such as research and education).
The cons:
1. Obviously more money flowing in means that locals who aren't wealthy will be squeezed out.
2. Gives foreigners a lot of influence and control over local policies.
3. Reduces supply of available housing.
The free-market way of thinking would be that supply will eventually catch up to demand. So if housing is expensive because there are more foreign buyers, then builders will be incentivized to build more housing, which in turn leads to lower prices as supply increases.
There are problems with this argument, however. It misses the fact that land itself is a limited commodity and that can't be free-marketed around. It also misses the fact that there are non-economic elements that the local population want to have. Things like clean air, preservation of historic buildings, cultural preservation (not wanting every building to look like a giant, generic, glass tower) and providing basic quality of life for every citizen regardless of economic circumstance.
That doesn't mean we need a knee-jerk ban on foreign investment, but it does mean the governments (local, state, maybe even federal) need to adjust public policies to strike a balance between economic forces (which have their advantages) and non-tangible desires of its citizens.
Some ideas:
1. As others have mentioned, tax unoccupied properties at a higher rate. Housing is meant for living in, not just a way to park cash. This doesn't necessarily discourage foreign investment but does make it so that people will be more willing to rent out (and thus provide a living space).
2. Allow more housing to be built. This won't solve the land problem but engaging creative architects and granting permits for contractors who not only want to build abundant housing but also build it in a way that preserves the general culture and feel of the location (something the citizens want) will at least provide more housing without demolishing what made the city unique.
3. Invest heavily in public transit and do it intelligently. Public transit is the one thing that can alleviate the land situation. It effectively increases the amount of livable land your citizens have access to. Allocate the money up-front instead of trickling it with stop-and-go budgets (governments love to do this) that end up costing more. Pick contractors based on performance. Make contracts fixed-cost and put heavy punitive measures for non-delivery.