For the record I think rent control is generally a very bad thing (one can make a case for much looser regulations which slow the rise of rents moderately , but let's ignore that for the moment). And, the record shows that rent control in New York (about the only place it exists at all) has been abysmal on both efficiency and equity grounds. That is, it was an extraordinarily expensive way to do very little to help people at the lower end of the economic spectrum.
But, Big Media Matt is quite right that in much of the country, including DC and New York, zoning regulations which prevent an increase in building heights have a much bigger impact on the (non)existence of relatively affordable housing. While rent control programs are an expensive way to do not too much for the poor, building height restrictions are an expensive way to do something (generally) for the relatively well off.
One could alleviate this problem while still having some respect for the very real issue of "neighborhood character." Walking around parts of New York a few weeks ago I saw plenty of examples where developers were allowed to build upwards and did so very poorly. Neighborhood character rarely has much to do with the existence or lack of tall buildings per se. They coexist rather nicely in my neighborhood for the most part. It is important to preserve street level access and retail, the original facade, the small "details" which make urban streetscapes interesting. This kind of thing can be done, though sadly it often isn't. For some reason local zoning boards rarely extract such concessions even though they're handing out a fairly valuable prize to developers. There are probably obvious reasons for that.
But, yes, generally - building up is the answer.
