Desertification would dry up agriculture and water sources the cities would depend on. Some cities (at a guess, those next to mountain ranges) would survive this because they could still tap rainwater from ocean evaporation. But it would stress (if not outright destroy) every city. It should be noted that not only is there less water to evaporate (thanks to all that ice), but the cooler world temperature would also lead to less evaporation. Water supply (for drinking water, sewage, and agriculture) would be a big problem.
Something else to consider, pipes are laid with an expectation of average soil humidity. Dry out the soil — threaten the pipe. Pipes carrying things like gasoline, fuel oil, natural gas, and who knows what nasty chemicals. Your large multi-leg power poles are probably fine, but your single-leg poles will rattle in their holes. And as the soil dries out, buildings will shift. Whole cities won't collapse, but individual buildings will develop cracks and require repair.
Every coastal city would become, basically, a non-coastal city. This isn't a deal-killer for the cities, but it will be very, very expensive. Cities located next to an underwater escarpment would need to dredge their harbors to regain access to the ocean. Those that didn't would need to rebuild their harbors. In that later case, city development would instantly shift from an inland-direction to a coastal-direction.
However, this is not only something expensive to repair, it would have devastating economic effects on the cities. Those with harbors have substantial dependencies on ocean trade — and it's not just the jobs at the piers that are effected. Every company that suddenly can't import/export through that harbor just went out of business. And that will have a domino effect that might not end even with the grade schools of dependent children or janitors. This has the ability to rewrite the entire socioeconomic structure of a city.
BTW, the same can be said for cities dependent on large rivers, even if they're inland cities. Most of those rivers would shrink or even dry up. An ice age can have dramatic effects. Chicago is as likely to suffer as Sydney or Hong Kong.
It's not obvious, but many industries must design manufacturing and processing plants to accommodate local humidity. And an ice age would change that. Consider your despair when Grandma Hansen's antique stand-up piano suddenly has a quarter-inch crack in it because the wood dried out. Now consider how you'll feel when your class-1 cleanroom suddenly isn't class-1 anymore because the filters and seals that expected Humidity have dried out. An ice age has the potential to take a lot of non-ocean-dependent industries offline until a tremendous price is paid to adjust to new climatological conditions.
And the secondary problems are too legion to address in a single question
Where do I begin with the chaos such an event could bring about? Large cities, more than any kind of community on Earth, are most susceptible to the proverbial "we're only one meal away from anarchy" idea. If the 1977 New York Blackout proved anything, it's that once you've lost control of a city's population, your one and only option is to let the situation burn itself out. A whole lot of dead people (and zombies, if modern entertainment has taught me anything, it's that there will always be zombies).
But, there's a saving grace
Movies like The Day After Tomorrow are fun, but in reality, ice ages don't occur in days, or even months. Humanity appears to be capable of heating things up,1 but we've also proven that we're not at all good at cooling things down. Thus, your ice age is most believably a consequence of natural events — and natural events take time.
Which means your cities have all the time in the world to accommodate the changes. Business will relocate, harbors will be dredged, etc. Quite frankly, while coastal cities in the situation you describe will have had to change with the climate — I don't believe it would have catastrophic effects at all. Agriculture would be relocated. Technologies invented. Yada-yada-yada. But, that's just me. I'm an optimist.
1 Over the course of a century.... I've been hearing "it's too late!" since the 70's, so I'm disinclined to believe it when I hear it today. But that's a discussion better suited for EarthScience.SE or Sustainability.SE, or even Outdoors.SE. ... or possibly Politics.SE or Skeptics.SE. Have you ever noticed there isn't a Weather.SE or Climate.SE? And yet we have an Alcohol.SE... humans will never change.